Codes with Weighted Poset Metrics Based on the Lattice of Subgroups of Z_m

Phichet Jitjankarn

doi:10.20944/preprints202410.1556.v1

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Codes with Weighted Poset Metrics Based on the Lattice of Subgroups of Z_m

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Phichet Jitjankarn^*

This version is not peer-reviewed.

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Submitted:

19 October 2024

Posted:

21 October 2024

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Abstract

By combining a lattice subgroup diagram of Z_m with a weighted poset metric, we introduce a new weighted coordinates poset metric for codes over Z_m , called LS-poset metric. When I is an ideal in a poset, the concept of I-perfect codes with LS-poset metric is investigated. We obtain a Singleton bound for codes with LS-poset metric and define MDS codes. When the poset in poset metric is a chain, we provide sufficient conditions for a code with LS-poset metric to be r-perfect for some r ∈ N .

Keywords:

MDS codes

;

poset codes

;

pomset codes

;

perfect codes

Subject:

Computer Science and Mathematics - Discrete Mathematics and Combinatorics

MSC: 94B05; 06A06; 15A03

1. Introduction

The concept of poset metric codes over a finite field

F_{q}

was introduced by Brualdi (see [2]) in 1995. Over the past two decades, coding theory has seen significant developments through the study of codes in the poset metric. This generalization of classical coding metrics has opened up new avenues for research and applications, particularly in scenarios where traditional metrics like the Hamming or Lee distance are not sufficient to model the complexities of error patterns. We refer to [1,3,5,6,7] for some results on poset metric spaces such as packing radius, the existence of r-error-correcting codes, perfect codes, and group of isometries. In 2018, the pomset metric was introduced by the authors in [8] to accommodate Lee metric for codes over

Z_{m}

. This metric is a further generalization of the poset metric and is based on the concept of pomsets, or partially ordered multisets. In both the poset and pomset metrics, the Singleton bound, MDS and I-perfect property for codes are studied (see [4,9]).

Both the poset and pomset metrics are constructed based on the structure of posets. The structure of a poset serves as the foundation for defining these metrics, as it establishes the relationships and dependencies between the elements of the codeword positions. In the present paper, we introduces a weighted poset metric based on subgroups diagram of

Z_{m}

. By using the poset of the power set of a multiset, we can effectively visualize the subgroup relationships in

Z_{m}

. The poset captures the inclusion relationships between subgroups, while the multiset represents the different ways subgroups can be generated based on the divisors of m. This approach is especially powerful for cyclic groups where the subgroup structure is tightly related to the divisors of the group’s order.

1.1. Poset Metrics

Let

P = ([n], ⪯_{P})

be a poset on the set

[n] : = {1, 2, \dots, n}

of coordinates of a vector in

F_{q}^{n}

(or

Z_{m}^{n}

). For

I \subseteq [n]

, I is called an (order) ideal of P if

i \in I

j ⪯_{P} i

imply that

j \in I

. For a subset S of P, we denote

〈 S 〉

the smallest ideal containing S. Given a vector

x = (x_{1}, \dots, x_{n}) \in F_{q}^{n}

, the support of

x

supp (x) = {i \in [n] : x_{i} \neq 0}

. The poset weight of

x

is defined as

w_{P} (x) : = | 〈 supp (x) 〉 | .

For

x, y \in F_{q}^{n}

, the poset distance between

x

and

y

d_{P} (x, y) : = w_{P} (x - y) .

It was shown in [2] that

d_{P}

is a metric on

F_{q}^{n}

. Notice that the poset metric

d_{P}

simplifies to the Hamming metric

d_{H}

when the poset P is an antichain.

1.2. Multisets and Pomsets

For given a nonempty set X and a map

c : X \to N_{0}

, an mset M is considered as a pair

M = (X, c)

. We write

a \in^{n} M

(or

n / a \in M

), if

c (a) \geq n > 0

, (i.e., a occurs in M at least n times).

An mset

M = (X, c)

, drawn from

X = {a_{1}, a_{2}, \dots, a_{t}}

, is represented as

M = {k_{1} / a_{1}, k_{2} / a_{2}, \dots, k_{t} / a_{t}},

where

c (a_{i}) = k_{i} > 0

for

i = 1, \dots, t

. If

k / a \in M

, then

r / a \in M

for all

1 \leq r \leq k

. The cardinality of an mset

M = (X, c)

is defined as

| M | = \sum_{x \in X} c (x)

For

m \in N

, we denote

M^{m} (X)

as the (regular) mset of height m drawn from the set X such that all elements of X occur with the same multiplicity m, i.e.,

| M^{m} (X) | = m | X |

. The mset space

{[X]}^{m}

is the set of all msets drawn from X such that no element in an mset occurs more than m times.

A submultiset (or submset) of an mset

M = (X, c)

is a multiset

S = (X, c_{S})

such that

c_{S} (x) \leq c (x)

for all

x \in X

. For an mset

M = (X, c)

, the set

M^{*} = {x \in X : c (x) > 0}

is called the root set of M.

For two msets

M_{1} = (X, c_{1})

and

M_{2} = (X, c_{2})

, we list some definitions of operatations in msets [8] as follows:

The addition(sum) of $M_{1}$ and $M_{2}$ is the mset $M_{1} \oplus M_{2} = (X, s)$ , where

$s (x) = c_{1} (x) + c_{2} (x)$ for all $x \in X$ .
The subtraction (difference) of $M_{1}$ from $M_{2}$ is the mset $M_{2} ⊖ M_{1} = (X, d)$ , where

$d (x) = max {c_{2} (x) - c_{1} (x), 0}$ for all $x \in X$ .
The union of $M_{1}$ and $M_{2}$ is the mset $M_{1} \cup M_{2} = (X, u)$ , where

$u (x) = max {c_{1} (x), c_{2} (x)}$ for all $x \in X$ .
The intersection of $M_{1}$ and $M_{2}$ is the mset $M_{1} \cap M_{2} = (X, i)$ , where

$i (x) = min {c_{1} (x), c_{2} (x)}$ for all $x \in X$ .

For

M_{1}, M_{2} \in {[X]}^{m}

, the mset sum

M_{1} \oplus M_{2} = (X, s) \in {[X]}^{m}

, where

s (x) = min {m, c_{M_{1}} (x) + c_{M_{2}} (x)}

for all

x \in X

. Given a submset

S = (X, b)

of an mset

M^{m} (X)

, the complement of S is an mset

S^{c} = (X, b_{c}) \in {[X]}^{m}

, where

b_{c} (x) = m - b (x)

for all

x \in X

For two msets

M_{1}, M_{2}

drawn fron a set X, we define the cartesian product

M_{1} \times M_{2}

M_{1} \times M_{2} : = {r s / (r / x, s / y) : r / x \in M_{1}, s / y \in M_{2}} .

A submset

R = (M \times M, g)

M \times M

is said to be an mset relation on M if

g (r / x, s / y) = r s

An mset relation R on M is called a partially ordered mset relation (or pomset relation) R on M if the following properties are all satisfied:

(1): [reflexivity] $\forall m / x \in M, (m / x, m / x) \in R$ ,
(2): [antisymmetry] if $(m / x, n / y), (n / y, m / x) \in R \Rightarrow m = n, x = y$ , and
(3): [transitivity] if $(m / x, n / y), (n / y, k / z) \in R \Rightarrow (m / x, k / z) \in R$ .

Notice that if

(m / x, m / x) \in R

, then

(r / x, s / x) \in R

for all

1 \leq r, s \leq m

For given a poset

P = (X, ⪯_{P})

, we define the pomset relation

≼_{P}

M^{m} (X)

having P-shape by

≼_{P} : = {m^{2} / (m / a, m / a), m^{2} / (m / a, m / b) : \forall a, b \in X, a ⪯_{P} b} .

The pair

(M^{m} (X), ≼_{P})

is known as a partially ordered multiset (pomset), denoted by

P

The dual pomset of the pomset

P = (M^{m} (X), ≼_{P})

, denoted by

\tilde{P}

, is the pomset on

M^{m} (X)

having

\tilde{P}

-shape, where

\tilde{P}

is the dual poset of P. That is,

m / a ≼_{P} m / b

P

if and only if

m / b ≼_{\tilde{P}} m / a

\tilde{P}

Let S be a submset of

M^{m} (X)

in a pomset

P = (M^{m} (X), ≼_{P})

. An element

t / a \in S

is said to be a maximal element in S if there is no element

k / c \in S (c \neq a)

such that

t / a ≼_{P} k / c

. An element

r / b \in S

is said to be a minimal element in S if there is no element

k / c \in S (c \neq b)

such that

k / c ≼_{P} t / b

Let

P = (M^{k} ([n]), ≼_{P})

be the pomset of height k having P-shape where the poset

P = ([n], ⪯_{P})

. An ideal in

P

is a submset

I \subseteq M^{k} ([n])

with the property that if

j / b \in I

and

i / a ≼_{P} j / b (a \neq b)

then

i / a \in I

. Given a submset S of

M^{k} ([n])

, we denote by

〈 S 〉

the smallest ideal containing S.

An ideal I of

M^{k} ([n])

is called an ideal with full count if

i / a \in I \Rightarrow k / a \in I

; otherwise, it is called an ideal with partial count.

Example 1.

From the poset

P_{1}

as in Figure 1, we consider the pomset

P_{1} = (M^{3} ([4]), ≼_{P_{1}})

. Let

I_{1} = {3 / 1, 3 / 4}

and

I_{2} = {3 / 1, 2 / 2, 1 / 3}

be ideals with full count and partial count in

P_{1}

, respectively. Then the complements

I_{1}^{c} = {3 / 2, 3 / 3}

and

I_{2}^{c} = {1 / 2, 2 / 3, 3 / 4}

are ideals with full count and partial count in

{\tilde{P}}_{1}

, respectively. Observe that

I_{1} \cap I_{1}^{c} = \emptyset

whereas

I_{2} \cap I_{2}^{c} = {1 / 2, 1 / 3}

Notice that if I is an ideal with full count in

P = (M^{k} ([n]), ≼_{P})

, then

{I, I^{c}}

is a partition of

M^{k} ([n])

, that is

I \cap I^{c} = \emptyset

and

I \cup I^{c} = M^{k} ([n])

. However, for given any submset J of

M^{k} ([n])

J \oplus J^{c} = M^{k} ([n])

1.3. Pomset Metrics

In the space

Z_{m}^{n}

with the pomset

P = (M^{⌊ m / 2 ⌋} ([n]), ≼_{P})

. For a vector

x = (x_{1}, \dots, x_{n}) \in Z_{m}^{n}

, the support of

x

with respect to Lee weight is defined to be

{supp}_{L} (x) : = {t / i : x_{i} \neq 0, and t = min {x_{i}, m - x_{i}}} .

The pomset weight of

x \in Z_{m}^{n}

is defined to be

w_{P} (x) : = | 〈 {supp}_{L} (x) 〉 |

, and the pomset distance between two vectors

x, y

Z_{m}^{n}

is defined by

d_{P} (x, y) : = w_{P} (x - y)

. It is known that the pomset distance is a metric on

Z_{m}^{n}

(see [8]), and it is called a pomset metric. When the pomset is an antichain, there is no hierarchical structure to influence the weight calculation, making the pomset metric equivalent to the Lee metric in terms of how the weight of the codeword is computed.

2. An Ordinal Product

Let

A = {n_{1} / a_{1}, n_{2} / a_{2}, \dots, n_{t} / a_{t}}

be an mset with

A^{*} = {a_{1}, a_{2}, \dots, a_{t}}

. Then

| A | = \sum_{i = 1}^{t} n_{i}

. Let

P (A)

be the power set of the mset A. With a slight change of notation, we will use

a_{i_{1}}^{[t_{i_{1}}]} a_{i_{2}}^{[t_{i_{2}}]} \dots a_{i_{s}}^{[t_{i_{s}}]}

for the mset

{t_{i_{1}} / a_{i_{1}}, t_{i_{2}} / a_{i_{2}}, \dots, t_{i_{s}} / a_{i_{s}}} \in P (A)

. Here we let

P (A) : = P (A) ∖ {\emptyset}

Note that we may write

a_{i}^{[0]}

to indicate that

a_{i}

does not appear. For

α_{1}, α_{2} \in P (A)

, which

α_{1} = a_{1}^{[l_{1}]} a_{2}^{[l_{2}]} \dots a_{t}^{[l_{t}]}

and

α_{2} = a_{1}^{[h_{1}]} a_{2}^{[h_{2}]} \dots a_{t}^{[h_{t}]}

, define the mset sum

α_{1} \oplus α_{2} = a_{1}^{[s_{1} (a_{1})]} a_{2}^{[s_{2} (a_{2})]} \dots a_{t}^{[s_{t} (a_{t})]} \in P (A)

, where

s_{i} (a_{i}) = min {n_{i}, l_{i} + h_{i}}

for all

i = 1, \dots, t

For each

α = a_{i_{1}}^{[t_{i_{1}}]} a_{i_{2}}^{[t_{i_{2}}]} \dots a_{i_{s}}^{[t_{i_{s}}]} \in P (A)

with

α \neq A

, the dual of

α

\hat{α} = a_{i_{1}}^{[n_{i_{1}} - t_{i_{1}}]} a_{i_{2}}^{[n_{i_{2}} - t_{i_{2}}]} \dots a_{i_{s}}^{[n_{i_{s}} - t_{i_{s}}]} \in P (A),

which

α \oplus \hat{α} = A

Under the submset relation

\subseteq_{A}

(P (A), \subseteq_{A})

is a partially ordered set, denoted by

P_{A}

. For each

α \in P (A)

, we denote

〈 α 〉

the ideal in

P (A)

having

α

as its maximum element. It is clear that

〈 A 〉 = P (A)

. For example, let

a^{[1]} b^{[2]}, b^{[3]} \in P (a^{[2]} b^{[3]})

. Then

〈 a^{[1]} b^{[2]} 〉 = {a^{[1]} b^{[2]}, a^{[1]} b^{[1]}, b^{[2]}, a^{[1]}, b^{[1]}}

, and

〈 b^{[3]} 〉 = {b^{[3]}, b^{[2]}, b^{[1]}}

Remark 1.

For

α = a_{i_{1}}^{[t_{i_{1}}]} a_{i_{2}}^{[t_{i_{2}}]} \dots a_{i_{s}}^{[t_{i_{s}}]} \in P (A)

, we have

| α | = \sum_{j = 1}^{s} t_{i_{j}}, and | 〈 α 〉 | = \prod_{j = 1}^{s} (t_{i_{j}} + 1) - 1

Given a poset

P = ([n], ⪯_{P})

, we define a relation

γ

[n] \times P (A)

(i, α) γ (j, β) \Leftrightarrow \{\begin{matrix} i = j and α \subseteq_{A} β \\ i ⪯_{P} j where i \neq j \end{matrix} .

It is clear that

([n] \times P (A), γ)

is a poset, denoted by

P \times P_{A}

. By the property of any ideal in a poset that contains every element smaller than or equal to some of its elements, we have that if

(i, α) \in I

and I is an ideal in

P \times P_{A}

, then

{i} \times 〈 α 〉 \subseteq I

An ideal I in

P \times P_{A} = ([n] \times P (A), γ)

is called an ideal with full count if

(i, α) \in I \Rightarrow (i, A) \in I

; otherwise, it is also called an ideal with partial count. Let

I (P \times P_{A})

be the set of all ideal in

P \times P_{A}

. For

I \in I (P \times P_{A})

, we denote

\begin{matrix} ω_{I_{f}} : = & {i \in [n] : (i, A) \in I}, and \\ ω_{I_{p}} : = & {i \in [n] : (i, A) \notin I but (i, α) \in I for some α \in P (A)} . \end{matrix}

Given an ideal with partial count I in

P \times P_{A}

and for

i \in ω_{I_{p}}

, we let

A (I; i) : = {α \in P (A) : (i, α) \in I}

. An ideal I in

P \times P_{A}

is called normal if

\forall i \in ω_{I_{p}}

A (I; i) = 〈 α_{i} 〉

for some

α_{i} \in P (A)

. We denote by

I (P \times P_{A})

the collections of normal ideals in

I (P \times P_{A})

The dual poset with respect toP of

P \times P_{A}

is the poset

\tilde{P} \times P_{A}

, where

\tilde{P}

is the dual poset of P. Let

I \in I (P \times P_{A})

. The complement of I, denoted by

I^{C}

, is a normal ideal in the dual poset

\tilde{P} \times P_{A}

which satisfies:

(i): $ω_{I_{p}^{C}} = ω_{I_{p}}$ and $ω_{I_{f}^{C}} = [n] ∖ (ω_{I_{f}} \cup ω_{I_{p}})$ , and
(ii): for $i \in ω_{I_{p}^{C}}$ , $A (I^{C}; i) = 〈 {\hat{α}}_{i} 〉$ , where $A (I; i) = 〈 α_{i} 〉$ for some $α_{i} \in P (A)$ .

Example 2.

Consider the poset

P_{3} = ([6], ⪯_{P_{3}})

as in Figure 1, and the mset

A = a^{[3]} b^{[2]}

. Let

I_{1}, I_{2}, I_{3} \in I (P_{3} \times P_{A})

be defined by

\begin{matrix} I_{1} & = {1, 2} \times 〈 a^{[3]} b^{[2]} 〉 \cup {4} \times 〈 a^{[2]} b^{[2]} 〉, \\ I_{2} & = {1} \times 〈 a^{[3]} b^{[2]} 〉, a n d I_{3} = {1} \times (〈 a^{[1]} b^{[2]} 〉 \cup 〈 a^{[3]} 〉) . \end{matrix}

It can see that

| I_{1} | = 30

| I_{2} | = 11

, and

| I_{3} | = 7

. We have

I_{1}

and

I_{3}

are ideals with partial count such that

I_{1} = 〈 (4, a^{[2]} b^{[2]}) 〉

and

I_{3} = 〈 {(1, a^{[1]} b^{[2]}), (1, a^{[3]})} 〉

, and

I_{2}

is an ideal with full count such that

I_{2} = 〈 (1, a^{[3]} b^{[2]}) 〉

. Clearly,

I_{1}

and

I_{2}

are normal, but

I_{3}

is not normal. For the complements of

I_{1}

and

I_{2}

, we have

\begin{matrix} I_{1}^{C} & = {3, 5, 6} \times 〈 a^{[3]} b^{[2]} 〉 \cup {(4, a^{[1]})}, a n d I_{2}^{C} & = {2, 3, 4, 5, 6} \times 〈 a^{[3]} b^{[2]} 〉 . \end{matrix}

For

α = a_{1}^{[h_{1}]} a_{2}^{[h_{2}]} \dots a_{t}^{[h_{t}]} \in P (A)

, where the mset

A = a_{1}^{[n_{1}]} a_{2}^{[n_{2}]} \dots a_{t}^{[n_{t}]}

and

0 \leq h_{i} \leq n_{i}

for

1 \leq i \leq t

, we define

⌊ α ⌉ : = \{\begin{matrix} | α |, & if h_{j} = n_{j} = 1 for some j \in {1, \dots, t}, \\ | α | + 1, & otherwise . \end{matrix}

The following result is directly obtained.

Proposition 1.

Given an mset

A = (A^{*}, c)

and

α \in P (A)

, we have that

1.: If $⌊ α ⌉ = | α | + 1 > 2$ , $\exists α^{'} \in P (A)$ such that $α^{'} \subseteq_{A} α$ and $⌊ α^{'} ⌉ = | α^{'} | + 1 = | α |$ .
2.: If $⌊ α ⌉ = | α | > 1$ , $\exists α^{'} \in P (A)$ such that $α^{'} \subseteq_{A} α$ and $⌊ α^{'} ⌉ = | α^{'} | = | α | - 1$ .
3.: For $α_{1} \subseteq_{A} α_{2} \in P (A)$ , if $⌊ α_{1} ⌉ = | α_{1} | + 1 = | α_{2} | = ⌊ α_{2} ⌉$ , then there is a unique $x \in α_{2}^{*} \subseteq A^{*}$ such that $c (x) = 1$ .
4.: If $c (x) > 1$ for all $x \in A^{*}$ , then there is no $α \in P (A)$ such that $⌊ α ⌉ = 1$ .

Define a map

ζ_{A} : I (P \times P_{A}) \to N

\begin{matrix} ζ_{A} (I) = & | ω_{I_{f}} | \cdot ⌊ A ⌉ + \sum_{\begin{matrix} i \in ω_{I_{p}} such that \\ A (I; i) = 〈 α_{i} 〉 for some α_{i} \in P (A) \end{matrix}} ⌊ α_{i} ⌉ . \end{matrix}

(1)

Observe that

ζ_{A} (I) \leq | I |

for all

I \in I (P \times P_{A})

. Moreover, if

| A | = 1

, then

ζ_{A} (I) = | I |

for all

I \in I (P \times P_{A})

. For

I \in I (P \times P_{A})

, let

\begin{matrix} Ω_{I_{p}} : = & {α \in P (A) : A (I; i) = 〈 α 〉 and ⌊ α ⌉ = | α | for i \in ω_{I_{p}}} and \\ {\bar{Ω}}_{I_{p}} : = & {α \in P (A) : A (I; i) = 〈 α 〉 and ⌊ α ⌉ = | α | + 1 for i \in ω_{I_{p}}} . \end{matrix}

Then

ζ_{A} (I) = | ω_{I_{f}} | \cdot ⌊ A ⌉ + [Ω_{I_{p}}]

, where

[Ω_{I_{p}}] = \sum_{α \in Ω_{I_{p}} \cup {\bar{Ω}}_{I_{p}}} ⌊ α ⌉

With Proposition 1 and by deleting a maximal element of a normal ideal I in

P \times P_{A}

, it gives a way to construct a normal ideal

J \subseteq I

. The next result is directly obtained.

Proposition 2.

Consider an mset

A = (A^{*}, c)

and the poset

P \times P_{A} = ([n] \times P (A), γ)

1.: For each $0 \leq t \leq n$ , there exists an ideal J with full count such that $ζ_{A} (J) = t \cdot ⌊ A ⌉$ .
2.: Let $I \in I (P \times P_{A})$ be such that $ω_{I_{p}} \neq \emptyset$ .

(2.1) If $Ω_{I_{p}} \neq \emptyset$ , then for each $0 \leq t \leq [Ω_{I_{p}}]$ there exists $J \in I (P \times P_{A})$ such

that $J \subseteq I$ and $ζ_{A} (J) = | ω_{I_{f}} | \cdot ⌊ A ⌉ + t$ .

(2.2) If $Ω_{I_{p}} = \emptyset$ , then for each $0 \leq t \leq [Ω_{I_{p}}]$ with $t \neq 1$ , there exists $J \in I (P \times P_{A})$

such that $J \subseteq I$ and $ζ_{A} (J) = | ω_{I_{f}} | \cdot ⌊ A ⌉ + t$ .

3. Supports and Weights

Suppose that

m = p_{1}^{β_{1}} p_{2}^{β_{2}} \dots p_{k}^{β_{k}}

, where

p_{1}, \dots, p_{k}

are distinct prime numbers and

β_{1}, \dots, β_{k}

are positive integers. By considering the mset

A = p_{1}^{[β_{1}]} p_{2}^{[β_{2}]} \dots p_{k}^{[β_{k}]}

, it can see that the lattice of subgroups of

Z_{m}

and the poset structure of

P (A)

under

\subseteq_{A}

are the same.

Let

Sub (Z_{m})

be the set of all subgroups of

Z_{m}

. The map

ψ : Sub (Z_{m}) \to P (A)

defined by

ψ (H) = α_{H} = p_{1}^{[t_{1}]} p_{2}^{[t_{2}]} \dots p_{k}^{[t_{k}]}

, where

| H | = p_{1}^{t_{1}} p_{2}^{t_{2}} \dots p_{k}^{t_{k}}

for

0 \leq t_{i} \leq β_{i}

, is an order-isomorphism.

Let

P = ([n], ⪯_{P})

be a poset. Given

x = (x_{1}, \dots, x_{n}) \in Z_{m}^{n}

, we define the support of x associated with the lattice of subgroups of

Z_{m}

{supp}_{L S} (x) : = {(i, p_{1}^{[t_{1}]} p_{2}^{[t_{2}]} \dots p_{k}^{[t_{k}]}) : x_{i} \neq 0 and | 〈 x_{i} 〉 | = p_{1}^{t_{1}} p_{2}^{t_{2}} \dots p_{k}^{t_{k}}}

a subset of

[n] \times P (A)

. By considering

〈 {supp}_{L S} (x) 〉 \in I (P \times P_{A})

as the smallest ideal in

P \times P_{A}

containing

{supp}_{L S} (x)

, the

L S

-poset weight of

x \in Z_{m}^{n}

is defined to be

w_{L S} (x) : = ζ_{A} (〈 {supp}_{L S} (x) 〉)

, and the

L S

-poset distance between

x, y \in Z_{m}^{n}

d_{L S} (x, y) : = w_{L S} (x - y)

. Now we prove that the

L S

-poset distance is a metric on

Z_{m}^{n}

Theorem 1.

Let

P = ([n], ⪯_{P})

be a poset and

m = p_{1}^{β_{1}} p_{2}^{β_{2}} \dots p_{k}^{β_{k}}

where

p_{i}

are distinct primes. Under the poset

P \times P_{A}

with the mset

A = p_{1}^{[β_{1}]} p_{2}^{[β_{2}]} \dots p_{k}^{[β_{k}]}

, the

L S

-poset distance

d_{L S} (\cdot, \cdot)

is a metric on

Z_{m}^{n}

Proof.

It is clear that

d_{L S} (x, y) \geq 0

. As a group

Z_{m}

, we have

d_{L S} (x, y) = 0

iff

x = y

. Moreover, for any

x \in Z_{m}

〈 x 〉 = 〈 - x 〉

, which implies that

d_{L S} (\cdot, \cdot)

is symmetric. To show that the triangle inequality of

d_{L S} (\cdot, \cdot)

holds, we let

a, b \in Z_{m}

. By applying the fundamental theorem of finite cyclic groups, we assume that

〈 a 〉 = 〈 p_{1}^{t_{1}} p_{2}^{t_{2}} \dots p_{k}^{t_{k}} 〉

and

〈 b 〉 = 〈 p_{1}^{s_{1}} p_{2}^{s_{2}} \dots p_{k}^{s_{k}} 〉

for some nonnegative integers

t_{i}, s_{i},

(1 \leq i \leq k)

. Then

| 〈 a 〉 | = p_{1}^{β_{1} - t_{1}} p_{2}^{β_{2} - t_{2}} \dots p_{k}^{β_{k} - t_{k}}

and

| 〈 b 〉 | = p_{1}^{β_{1} - s_{1}} p_{2}^{β_{2} - s_{2}} \dots p_{k}^{β_{k} - s_{k}}

. Suppose that

〈 a + b 〉 = 〈 p_{1}^{r_{1}} p_{2}^{r_{2}} \dots p_{k}^{r_{k}} 〉

for some

r_{i} \geq 0

. It is clear that

〈 a + b 〉 \subseteq H = 〈 p_{1}^{min {t_{1}, s_{1}}} p_{2}^{min {t_{2}, s_{2}}} \dots p_{k}^{min {t_{k}, s_{k}}} 〉

. Then

α_{〈 a + b 〉} = p_{1}^{[β_{1} - r_{1}]} p_{2}^{[β_{2} - r_{2}]} \dots p_{k}^{[β_{k} - r_{k}]} \subseteq_{A} α_{H} = p_{1}^{[β_{1} - min {t_{1}, s_{1}}]} p_{2}^{[β_{2} - min {t_{2}, s_{2}}]} \dots p_{k}^{[β_{k} - min {t_{k}, s_{k}}]}

. Observe that

| α_{〈 a + b 〉} | = \sum_{i = 1}^{k} (β_{i} - r_{i}) \leq \sum_{i = 1}^{k} [β_{i} - min {t_{i}, s_{i}}] = | α_{H} | \leq \sum_{i = 1}^{k} [(β_{i} - t_{i}) + (β_{i} - s_{i})] = | α_{〈 a 〉} | + | α_{〈 b 〉} |

. If

| α_{〈 a + b 〉} | < | α_{H} |

, it follows that

⌊ α_{〈 a + b 〉} ⌉ \leq ⌊ α_{〈 a 〉} ⌉ + ⌊ α_{〈 b 〉} ⌉

. Now, we suppose

| α_{〈 a + b 〉} | = | α_{H} |

. From

α_{〈 a + b 〉} \subseteq_{A} α_{H}

, this forces

α_{〈 a + b 〉} = α_{H}

which means

r_{i} = min {t_{i}, s_{i}}

for all i. It is clear for the case

⌊ α_{〈 a + b 〉} ⌉ = | α_{〈 a + b 〉} |

. Next, assume that

⌊ α_{〈 a + b 〉} ⌉ = | α_{〈 a + b 〉} | + 1

. It follows that if

β_{j} = 1

for

j \in {1, \dots, k}

, then

1 = r_{j} = min {t_{j}, s_{j}}

which implies

⌊ α_{〈 a 〉} ⌉ = | α_{〈 a 〉} | + 1

and

⌊ α_{〈 b 〉} ⌉ = | α_{〈 b 〉} | + 1

. Consequently,

⌊ α_{〈 a + b 〉} ⌉ \leq ⌊ α_{〈 a 〉} ⌉ + ⌊ α_{〈 b 〉} ⌉

. This completes the proof. □

The metric

d_{L S} (\cdot, \cdot)

Z_{m}^{n}

is called as the

L S

-poset metric. Let C be a submodule of

Z_{m}^{n}

with the

L S

-poset metric

d_{L S}

. Then C is called an

L S

-poset code of length n over

Z_{m}

. The minimum

L S

-poset distance

d_{L S} (C)

is the smallest

L S

-poset distance between two distinct codewords of C. The dual of an

L S

-poset code C is defined as

C^{⊥} = {v \in Z_{m}^{n} | u \cdot v = u_{1} v_{1} + u_{2} v_{2} + \dots + u_{n} v_{n} = 0 for all u \in C} .

To obtain more information on each element of

Z_{m} ∖ {0}

which is placed on the poset structure of

P (A)

, we let

G_{t} : = {x \in Z_{m} ∖ {0} | ⌊ α_{〈 x 〉} ⌉ = t} .

Example 3.

Consider

Z_{20}

with the mset

A = 2^{[2]} 5^{[1]}

. We have the following table:

$P (A)$	$⌊ \cdot ⌉$	$Z_{20} ∖ {0}$
$5^{[1]}$	1	$4, 8, 12, 16$
$2^{[1]}$	2	10
$2^{[1]} 5^{[1]}$	2	$2, 6, 14, 18$
$2^{[2]}$	3	$5, 15$
$2^{[2]} 5^{[1]}$	3	$1, 3, 7, 9, 11, 13, 17, 19$

Recall some properties of the Euler

ϕ

-function as follows:

If p is a prime, then $ϕ (p) = p - 1$ and $ϕ (p^{k}) = p^{k} - p^{k - 1}$ for all $k \in N$ .
For $x, y \in N$ , if $gcd (x, y) = 1$ , then $ϕ (x y) = ϕ (x) ϕ (y)$ .

Remark 2.

Suppose

m = p_{1}^{β_{1}} p_{2}^{β_{2}} \dots p_{k}^{β_{k}}

, where

p_{i}

are distinct primes. Consider the mset

A = p_{1}^{[β_{1}]} p_{2}^{[β_{2}]} \dots p_{k}^{[β_{k}]}

. We have that

If $β_{i} > 1$ for all i, then $G_{1} = \emptyset$ and $| G_{| A | + 1} | = ϕ (m)$ .
If $β_{i} = 1$ for all i, then for $1 \leq t \leq k$ ,

$| G_{t} | = \sum_{\begin{matrix} (s_{1}, \dots, s_{k}) where \\ t = \sum_{i}^{k} s_{i}, s_{i} \in {0, 1} \end{matrix}} ϕ (p_{1}^{s_{1}} p_{2}^{s_{2}} \dots p_{k}^{s_{k}}) .$

Example 4.

In the space

Z_{180}^{4}

, we consider the poset

P_{1} \times P_{A} = ([4] \times P (A), γ)

where the mset

A = 2^{[2]} 3^{[2]} 5^{[1]}

and the poset

P_{1} = ([4], ⪯_{P_{1}})

is as shown in Figure 1. Consider the vector

(0, 5, 12, 120) \in Z_{180}^{4}

. We have

(Poset weight) $w_{P_{1}} ((0, 5, 12, 120)) = | 〈 {2, 3, 4} 〉 | = 4$ .
(Pomset weight) $w_{P_{1}} ((0, 5, 12, 120)) = | 〈 {5 / 2, 12 / 3, 60 / 4} 〉 | = 77 .$
( $L S$ -poset weight)

$w_{L S} ((0, 5, 12, 120)) = ζ_{A} 〈 {(2, 2^{[2]} 3^{[2]}), (3, 3^{[1]} 5^{[1]}), (4, 3^{[1]})} 〉 = 14 .$

Notice that for a space

Z_{p}^{n}

with prime p, the poset metric

d_{P}

and the

L S

-poset metric

d_{L S}

are the same, while the pomset metric

d_{P}

and the

L S

-poset metric

d_{L S}

are equivalent when

p = 2, 3

. The diagram in Figure 2 illustrates these facts.

4. r-Balls and I-Balls

Let

u

be a vector in the space

Z_{m}^{n}

with

L S

-poset metric

d_{L S}

and

r \in N_{0}

. With center at

u

and radius r, the r-ball and the r-sphere, respectively, are as follows:

\begin{matrix} B_{r, L S} (u) & : = {v \in Z_{m}^{n} | d_{L S} (u, v) \leq r}, \\ S_{r, L S} (u) & : = {v \in Z_{m}^{n} | d_{L S} (u, v) = r} . \end{matrix}

It is clear that

| B_{r, L S} (u) | = 1 + \sum_{i = 1}^{r} | S_{i, L S} (u) |

Definition 1.

Let C be a code of

Z_{m}^{n}

with

L S

-poset metric

d_{L S}

. Then C is said to be a r-perfect

L S

-poset code if the r-balls centered at the codewords of C are pairwise disjoint and their union is

Z_{m}^{n}

Let

I^{t} (s) \in I (P \times P_{A})

be such that

ζ_{A} (I^{t} (s)) = t

and

I^{t} (s)

has exactly s maximal elements. We let

ω_{I^{t} (s)} = {i \in [n] | (i, α) is a maximal element of I^{t} (s) for some α \in P (A)}

. Given a vector

v = (v_{1}, \dots, v_{n})

Z_{m}^{n}

, we rewrite it as

v : = (v_{1} : v_{2} : 0) = ({v_{i}}_{i \in ω_{I^{t} (s)}} : {v_{i}}_{i \in I^{t} {(s)}^{*} ∖ ω_{I^{t} (s)}} : 0),

where for each

i \in [n]

v_{i}

is an element in

Z_{m}

satisfying:

(i): If $i \in ω_{I^{t} (s)}$ , $v_{i} \in G_{⌊ α_{i} ⌉}$ ,

where $A (I^{t} (s); i) = 〈 α_{i} 〉$ for some $α_{i} \in P (A)$ ;
(ii): If $i \in I^{t} {(s)}^{*} ∖ ω_{I^{t} (s)}$ , $v_{i} \in Z_{m}$ ;
(iii): If $i \notin I^{t} {(s)}^{*}$ , $v_{i} = 0$ .

Observe that

〈 {supp}_{L S} (v) 〉 = I^{t} (s)

. Now, letting

A_{I^{t} (s)}

the collection of all vectors

v

Z_{m}^{n}

such that

〈 {supp}_{L S} (v) 〉 = I^{t} (s)

, we have

| A_{I^{t} (s)} | = m^{| I^{t} {(s)}^{*} ∖ ω_{I^{t} (s)} |} \cdot \prod_{i \in ω_{I^{t} (s)}} | G_{⌊ α_{i} ⌉} | .

Obviously, for two distinct ideals I and J in

I (P \times P_{A})

A_{I} \cap A_{J} = \emptyset

. Now, we denote by

I^{t} (s)

the set of all ideals

I \in I (P \times P_{A})

such that

ζ_{A} (I) = t

, and I has exactly s maximal elements. Then the number of vectors in an r-ball with center

u

equals

| B_{r, L S} (u) | = 1 + \sum_{i = 1}^{r} \sum_{j = 1}^{i} \sum_{\begin{matrix} I \in I^{i} (j) \neq \emptyset \end{matrix}} | A_{I} | .

(2)

Given an ideal

I \in I (P \times P_{A})

, the I-ball centered at

u

and the I-sphere centered at

u

, respectively, are defined as

\begin{matrix} B_{I, L S} (u) & : = {v \in Z_{m}^{n} | 〈 {supp}_{L S} (u - v) 〉 \subseteq I}, \\ S_{I, L S} (u) & : = {v \in Z_{m}^{n} | 〈 {supp}_{L S} (u - v) 〉 = I} . \end{matrix}

Definition 2.

Let C be a code of

Z_{m}^{n}

with

L S

-poset metric

d_{L S}

and I be an ideal in

P \times P_{A}

. Then C is called an I-perfect

L S

-poset code if the I-balls centered at the codewords of C are pairwise disjoint and their union is

Z_{m}^{n}

Under pomset metric

d_{P}

Z_{m}^{n}

, it was shown in [9] that I-balls are no more linear subspaces of

Z_{m}^{n}

if I is an ideal with partial count in

P = (M^{⌊ m / 2 ⌋} ([n]), ≼_{P})

. On the other hand, with

L S

-poset metric, the I-ball centered at the zero vector is a submodule of

Z_{m}^{n}

Proposition 3.

Let

I \in I (P \times P_{A})

. Then

B_{I, L S} (0)

is a submodule of

Z_{m}^{n}

Proof.

Clearly, if I is an ideal with full count in

P \times P_{A}

, then

B_{I, L S} (0)

is a submodule of

Z_{m}^{n}

with dimension

| ω_{I_{f}} |

. Now, suppose that I is an ideal with partial count. Then

| ω_{I_{p}} | > 0

. For each

i \in ω_{I_{p}}

, let

A (I; i) = 〈 α_{i} 〉

for some

α_{i} \in P (A)

. By considering

H_{i}

as a subgroup of

Z_{m}

such that

ψ (H_{i}) = α_{i}

for

i \in ω_{I_{p}}

, for

u = (u_{1}, \dots, u_{n}), v = (v_{1}, \dots, v_{n}) \in B_{I, L S} (0)

, we have

u_{i}, v_{i} \in H_{i}

for all

i \in ω_{I_{p}}

. It follows that

u + v, c u \in B_{I, L S} (0)

for

c \in Z_{m}

. Hence,

B_{I, L S} (0)

is a submodule of

Z_{m}^{n}

. □

For

I \in I (P \times P_{A})

, let

B_{I^{C}, \tilde{L S}} (0)

denote the

I^{C}

-ball centered at

0

under the poset

\tilde{P} \times P_{A}

Proposition 4.

Let

I \in I (P \times P_{A})

. Then the following statements hold:

1.: For $u \in Z_{m}^{n}$ , $B_{I, L S} (u) = u + B_{I, L S} (0)$ .
2.: For $u, v \in Z_{m}^{n}$ , $B_{I, L S} (u)$ and $B_{I, L S} (v)$ are either identical or disjoint. Moreover,

$B_{I, L S} (u) = B_{I, L S} (v) \Leftrightarrow {supp}_{L S} (u - v) \subseteq I$ .
3.: $B_{I, L S}^{⊥} (0) = B_{I^{C}, \tilde{L S}} (0)$ .

Proof.

(1) Let

v \in B_{I, L S} (u)

. It follows that

u - v \in B_{I, L S} (0)

, and

v = u + (v - u) \in u + B_{I, P_{C}} (0)

. For

w \in B_{I, P_{C}} (0)

, we have

{supp}_{L S} (u - (u + w)) = {supp}_{L S} (- w) = {supp}_{L S} (w) \subseteq I

. Hence

u + w \in B_{I, L S} (u)

(2) For each

i \in ω_{I_{f}} \cup ω_{I_{p}}

, we let

H_{i}

be a subgroup of

Z_{m}

such that

ψ (H_{i}) = α_{H_{i}} \in P (A)

, where

A (I; i) = 〈 α_{H_{i}} 〉

. For

u = (u_{1}, \dots, u_{n}), v = (v_{1}, \dots, v_{n}) \in Z_{m}^{n}

, suppose

w = (w_{1}, \dots, ω_{n}) \in B_{I, L S} (u) \cap B_{I, L S} (v)

. We have

{supp}_{L S} (u - w) \subseteq I

and

{supp}_{L S} (v - w) \subseteq I

. If

i \notin ω_{I_{f}} \cup ω_{I_{p}}

, then

u_{i} - w_{i} = w_{i} - v_{i} = 0

, so

u_{i} - v_{i} = 0

. For the case

i \in ω_{I_{f}} \cup ω_{I_{p}}

, we have

u_{i} - w_{i}, w_{i} - v_{i} \in H_{i}

, which implies

u_{i} - v_{i} \in H_{i}

. That is,

(i, α_{〈 u_{i} - v_{i} 〉}) \in I

for all

i \in ω_{I_{f}} \cup ω_{I_{p}}

. Consequently,

u - v \in B_{I, L S} (0)

, which means that

B_{I, L S} (u) = B_{I, L S} (v)

(3) If I is an ideal with full count in

P \times P_{A}

, then

I^{C}

is also an ideal with full count in

\tilde{P} \times P_{A}

. Since

ω_{I_{f}} \cap ω_{I_{f}^{C}} = \emptyset

and

ω_{I_{f}} \cup ω_{I_{f}^{C}} = [n]

, we derive the result.

Next, suppose that

m = p_{1}^{β_{1}} p_{2}^{β_{2}} \dots p_{k}^{β_{k}}

, where

p_{i}

are distinct primes, and the mset

A = p_{1}^{[β_{1}]} p_{2}^{[β_{2}]} \dots p_{k}^{[β_{k}]}

. Let

I \in I (P \times P_{A})

be an ideal with partial count. From

ω_{I_{p}} = ω_{I_{p}^{C}}

, for each

i \in ω_{I_{p}}

, we let

A (I; i) = 〈 α_{i} 〉

for some

α_{i} \in P (A)

, where

α_{i} = p_{1}^{[t_{i_{1}}]} p_{2}^{[t_{i_{2}}]} \dots p_{k}^{[t_{i_{k}}]}

, and let

H_{i}

and

K_{i}

be subgroups of

Z_{m}

such that

ψ (H_{i}) = α_{i}

and

ψ (K_{i}) = \hat{α_{i}}

. Then

H_{i} = 〈 p_{1}^{β_{1} - t_{i_{1}}} p_{2}^{β_{2} - t_{i_{2}}} \dots p_{k}^{β_{k} - t_{i_{k}}} 〉

and

K_{i} = 〈 p_{1}^{t_{i_{1}}} p_{2}^{t_{i_{2}}} \dots p_{k}^{t_{i_{k}}} 〉

. Let

x = (x_{1}, \dots, x_{n}) \in B_{I^{C}, \tilde{L S}} (0)

and

y = (y_{1}, \dots, y_{n}) \in B_{I, L S} (0)

. Then

x \cdot y = \sum_{i \in ω_{I_{p}}} x_{i} y_{i}

, where for each i,

x_{i} \in H_{i}

and

y_{i} \in K_{i}

. It follows that

x \cdot y

is congruent to 0 modulo m. That is,

B_{I^{C}, \tilde{L S}} (0) \subseteq B_{I, L S}^{⊥} (0)

. Now, we assume that there is

z \in B_{I, L S}^{⊥} (0) ∖ B_{I^{C}, \tilde{L S}} (0)

. Since

z \cdot y = 0

for all

y \in B_{I, L S} (0)

, we can, without loss of generality, write

z = (0, \dots, 0, z_{s}, 0, \dots, 0)

, where

s \in ω_{I_{p}}

and

z_{s} \notin K_{s} = 〈 p_{1}^{t_{s_{1}}} p_{2}^{t_{s_{2}}} \dots p_{k}^{t_{s_{k}}} 〉

. Then

z_{s} = y p_{1}^{r_{s_{1}}} p_{2}^{r_{s_{2}}} \dots p_{k}^{r_{s_{k}}}

, where

gcd (y, p_{j}) = 1

and

0 \leq r_{s_{j}} < t_{s_{j}}

for some

j \in [k]

. Choosing

w = (w_{1}, \dots, w_{n}) \in B_{I, L S} (0)

, defined by

w_{i} = 0

for all

i \neq s

, and

w_{s} = p_{1}^{β_{1} - t_{s_{1}}} p_{2}^{β_{2} - t_{s_{2}}} \dots p_{k}^{β_{k} - t_{s_{k}}}

, it follows that

z \cdot w \neg \equiv 0

modulo m, which is a contradiction. □

Example 5.

Consider

120 = 2^{3} \cdot 3 \cdot 5

and the mset

A = 2^{[3]} 3^{[1]} 5^{[1]}

. On

P (A)

, we choose

α_{1} = 2^{[1]} 3^{[1]} 5^{[1]}, α_{2} = 2^{[3]} 5^{[1]}, α_{3} = 2^{[2]} 3^{[1]}

. The structure of each

〈 α_{i} 〉

when

i = 1, 2, 3

, is demonstrated via the lattice of nontrivial subgroup for

Z_{120}

(see in Figure 3) in which

{\hat{α}}_{1} = 2^{[2]}

{\hat{α}}_{2} = 3^{[1]}

, and

{\hat{α}}_{3} = 2^{[1]} 5^{[1]}

. Now let us consider the poset

P_{1} \times P_{A} = ([4] \times P (A), γ)

, where the poset

P_{1} = ([4], ⪯_{P_{1}})

is as shown in Figure 1. Let

I = 〈 {(2, α_{1}), (3, α_{2}), (4, α_{3})} 〉 \in I (P_{1} \times P_{A})

. Then I is an ideal with partial count. It is easy to see that

B_{I, L S} (0) = Z_{120} \times 〈 4 〉 \times 〈 3 〉 \times 〈 10 〉

, whereas

B_{I, L S}^{⊥} (0) = 〈 0 〉 \times 〈 30 〉 \times 〈 40 〉 \times 〈 12 〉

Observe that the I-ball centered at the zero vector can be considered as a direct product of cyclic subgroups of

Z_{m}

. If m is a prime power, the following result is directly obtained.

Proposition 5.

In the space

Z_{q^{β}}^{n}

, let

I \in I (P \times P_{A})

.Then

1.: If $β = 1$ , then $| B_{I, L S} (0) | = q^{| I |}$ and $| B_{I, L S}^{⊥} (0) | = q^{n - | I |}$ .
2.: If $β > 1$ , then

$| B_{I, L S} (0) | = q^{β | ω_{I_{f}} | + \sum_{i \in ω_{I_{p}}} | α_{i} |}$ and $| B_{I, L S}^{⊥} (0) | = q^{β (n - | ω_{I_{f}} \cup ω_{I_{p}} |) + \sum_{i \in ω_{I_{p}}} | {\hat{α}}_{i} |}$ ,

where for each $i \in ω_{I_{p}}$ , $A (I; i) = 〈 α_{i} 〉$ for some $α_{i} \in P (A)$ .

From Proposition 3 and 4, the following theorem shows the existence of an I-perfect code with

L S

-poset metric when I is an ideal with full count.

Theorem 2.

For any ideal I with full count in

P \times P_{A}

, we have

1.: $B_{I^{C}, \tilde{L S}} (0)$ is an I-perfect $L S$ -poset code for the poset $P \times P_{A}$ .
2.: $B_{I, L S} (0)$ is an $I^{C}$ -perfect $L S$ -poset code for the poset $\tilde{P} \times P_{A}$ .

In the case of ideals I with partial count, the I-ball centered at the zero vector is not always I-perfect. The next lemma is a key for the existence of I-perfect code with

L S

-poset metric.

For each

i \in [n]

, let

e_{i} = (e_{1}, \dots, e_{n}) \in Z_{m}^{n}

be such that

e_{j} = 0

j \neq i

, and

e_{i} = 1

Lemma 1.

Let

I \in I (P \times P_{A})

with

ω_{I_{p}} \neq \emptyset

. For each

i \in ω_{I_{p}}

, let

H_{i}

be a nontrivial subgroup of

Z_{m}

such that

A (I; i) = 〈 α_{H_{i}} 〉

. Then the following statements hold:

1.: If there is $i \in ω_{I_{p}}$ such that $α_{H_{i}}^{*} = A^{*}$ , then there is no I-perfect $L S$ -poset code.
2.: Suppose C is an I-perfect $L S$ -poset code of $Z_{m}^{n}$ . Then for each $i \in ω_{I_{p}}$ , there is a maximal subgroup $K_{i}$ of $Z_{m}$ such that $A^{*} = α_{K_{i}}^{*} \dot{\cup} α_{H_{i}}^{*}$ . Moreover, $e_{i} \cdot C = K_{i}$ and $K_{i} + H_{i} = Z_{m}$ .

Proof.

(1) Suppose that

α_{H_{i}}^{*} = A^{*}

. Choose

v = (0, \dots, 0, v_{i}, 0, \dots, 0) \in Z_{m}^{n}

, where

v_{i} \notin H_{i}

. Then

α_{〈 v_{i} 〉} \notin 〈 α_{H_{i}} 〉

. It follows that

v \notin B_{I, L S} (0)

. Suppose there is an I-perfect

L S

-poset code C of

Z_{m}^{n}

. Then

v \in B_{I, L S} (c)

for some

c = (c_{1}, \dots, c_{n}) \in C

. That is,

{supp}_{L S} (c - v) \subseteq I

. This implies that

α_{〈 c_{i} - v_{i} 〉} \in 〈 α_{H_{i}} 〉

which means

0 \neq c_{i} \in v_{i} + H_{i}

. From

α_{H_{i}}^{*} = A^{*}

, there is

N_{0} \in N

such that

N_{0} c_{i} \neq 0

and

α_{〈 N_{0} c_{i} 〉} \in 〈 α_{H_{i}} 〉

. Consequently,

{supp}_{L S} (N_{0} c) \subseteq I

. As a submodule

B_{I, L S} (0)

Z_{m}^{n}

, we have

0 \neq N_{0} c \in B_{I, L S} (0)

, which is a contradiction to the I-perfect of C.

(2) Suppose C is an I-perfect

L S

-poset code of

Z_{m}^{n}

. Let

i \in ω_{I_{p}}

. From (1),

α_{H_{i}}^{*} \neq A^{*}

. Then there is a maximal subgroup

K_{i}

Z_{m}

such that

α_{K_{i}}^{*} \cap α_{H_{i}}^{*} = \emptyset

. Let

0 \neq x \in K_{i}

. Consider

v = (0, \dots, 0, v_{i} = x, 0, \dots, 0) \in Z_{m}^{n}

. Then

v \notin B_{I, L S} (0)

. We choose

c = (c_{1}, \dots, c_{n}) \in C

such that

v \in B_{I, L S} (c)

. Since

B_{I, L S} (0) \cap B_{I, L S} (c) = \emptyset

, it follows that

c_{i} \neq 0

and

α_{〈 c_{i} 〉} \notin 〈 α_{H_{i}} 〉

. Indeed, by proceeding as before, we have

α_{〈 c_{i} 〉}^{*} \cap α_{H_{i}}^{*} = \emptyset

. That is,

〈 c_{i} 〉 \cap H_{i} = {0}

. Since

{supp}_{L S} (c - v) \subseteq I

, we have

α_{〈 c_{i} - x 〉} \in 〈 α_{H_{i}} 〉

which means

〈 c_{i} - x 〉 \subseteq H_{i}

. Then

c_{i} - x = y

for some

y \in H_{i}

. From

α_{K_{i}}^{*} \cap α_{H_{i}}^{*} = \emptyset

, we have

K_{i} \cap H_{i} = {0}

. Then there is

N \in N

such that

N ∤ | H_{i} |

and

N x \equiv 0

modulo m. Thus,

N c_{i} \equiv N y

modulo m. These force

y = 0

. Hence,

c_{i} = x

. That is,

e_{i} \cdot C = K_{i}

. Since C is I-perfect, by a similar technique, it can be shown that

K_{i} + H_{i} = Z_{m}

. □

Given an mset A, let

E_{A} : = {α \in P (A) | α^{*} \cap {\hat{α}}^{*} = \emptyset}

Theorem 3.

Let

I \in I (P \times P_{A})

with

ω_{I_{p}} \neq \emptyset

. Then

B_{I, L S}^{⊥} (0)

is an I-perfect

L S

-poset code of

Z_{m}^{n}

if and only if for each

i \in ω_{I_{p}}

A (I; i) = 〈 α_{i} 〉

for some

α_{i} \in E_{A}

Proof.

Suppose

B_{I, L S}^{⊥} (0)

is I-perfect. By Proposition 4(3),

B_{I, L S}^{⊥} (0) = B_{I^{C}, \tilde{L S}} (0)

. For the necessary condition, let

i \in ω_{I_{p}}

. We have

A (I; i) = 〈 α_{H_{i}} 〉

and

A (I^{C}; i) = 〈 {\hat{α}}_{H_{i}} 〉 = 〈 α_{K_{i}} 〉

, where two subgroups

H_{i}, K_{i}

are as in Lemma 1. Then

α_{H_{i}} \in E_{A}

For each

α \in E_{A}

, we have that

α^{*} \cap {\hat{α}}^{*} = \emptyset

and

α \oplus \hat{α} = A

. These imply that

ψ^{- 1} (α) \cap ψ^{- 1} (\hat{α}) = {0}

and

| ψ^{- 1} (α) | \cdot | ψ^{- 1} (\hat{α}) | = m

. Hence, the converse is proved. □

Corollary 1.

There is no I-perfect

L S

-poset code of

Z_{q^{β}}^{n}

if I is an ideal with partial count in

P \times P_{A}

Example 6.

In Example 5, we have

α_{2} = 2^{[3]} 5^{[1]} \in E_{A}

with

{\hat{α}}_{2} = 3^{[1]}

, but

α_{1}, α_{3} \notin E_{A}

. Consider the poset

P_{2} = ([6], ⪯_{P_{2}})

as in Figure 1. In the space

Z_{120}^{6}

with

L S

-poset metric, we let

I = 〈 (4, 2^{[3]} 5^{[1]}) 〉 \in I (P_{2} \times P_{A})

as an ideal with partial count. By Theorem 3, we have

B_{I, L S}^{⊥} (0) = {(0, 0, 0, x, y, z) | x \in 〈 40 〉 = ψ^{- 1} (3^{[1]}), and y, z \in Z_{120}}

is an I-perfect

L S

-poset code of

Z_{120}^{6}

5. MDS $L S$ -Poset Codes and Codes in Chain Poset Structure

Theorem 4.

(Singleton Bound) Let

P \times P_{A}

be the poset on

[n] \times P (A)

and

C \subseteq Z_{m}^{n}

be an

L S

-poset code. Then

\begin{matrix} {log}_{m} | C | & \leq n - ⌊\frac{d_{L S} (C) - 1}{⌊ A ⌉}⌋ \end{matrix}

(3)

Proof.

Choose

x, y \in C

such that

d_{L S} (x, y) = d_{L S} (C)

. Consider the ideal I generated by

{supp}_{L S} (x - y)

. We have

d_{L S} (C) - 1 < ζ_{A} (I) \leq ⌊ A ⌉ \cdot | ω_{I_{f}} \cup ω_{I_{p}} |

. From Proposition 2, there is a normal ideal J with full count of

P \times P_{A}

such that

| ω_{J_{f}} | = ⌊\frac{d_{L S} (C) - 1}{⌊ A ⌉}⌋

. Then

ζ_{A} (J) = ⌊\frac{d_{L S} (C) - 1}{⌊ A ⌉}⌋ ⌊ A ⌉ \leq d_{L S} (C) - 1

. That is, there is no codeword

c

in C such that

{supp}_{L S} (c) \subseteq J

, and any two distinct codewords of C will not coincide in all position

j \in [n] ∖ ω_{J_{f}}

. These imply that

| C | \leq m^{n - | ω_{J_{f}} |}

. So we have

{log}_{m} | C | \leq n - ⌊\frac{d_{L S} (C) - 1}{⌊ A ⌉}⌋

. □

Definition 3.

L S

-poset code C of length n over

Z_{m}

is said to be amaximum distance separable

L S

-poset code(or simply MDS

L S

-poset code) if it attains the Singleton bound as in (3).

Theorem 5.

In the space

Z_{m}^{n}

with the poset

P \times P_{A}

, let

C \subseteq Z_{m}^{n}

be an

L S

-poset code such that

| C | = m^{t}

. Then C is an MDS

L S

-poset code if and only if C is an I-perfect

L S

-poset code for all ideals

I \in I (P \times P_{A})

with full count such that

| ω_{I_{f}} | = n - t

Proof.

Let I be an ideal with full count in

P \times P_{A}

such that

| ω_{I_{f}} | = n - t

. It is clear that

| B_{I, L S} (0) | = m^{n - t}

. Suppose that C is an MDS

L S

-poset code. From (3), it follows that

⌊\frac{d_{L S} (C) - 1}{⌊ A ⌉}⌋ = n - t

. Then

d_{L S} (C) > ζ_{A} (I)

, that is,

B_{I, L S} (x) \cap B_{I, L S} (y) = \emptyset

for two distinct elements

x, y \in C

. By

| C | = m^{t}

, we have C is I-perfect.

To show that C is MDS, we choose

c \in C ∖ {0}

such that

w_{L S} (c) = d_{L S} (C)

. Let

J = 〈 {supp}_{L S} (c) 〉

. Suppose

ζ_{A} (J) \leq ⌊ A ⌉ (n - t)

. By Proposition 2, we can construct an ideal I with full count in

P \times P_{A}

containing J such that

ζ_{A} (I) = ⌊ A ⌉ (n - t)

. But this would imply that

c \in B_{I, L S} (0)

, which is impossible since C is I-perfect. This forces that

d_{L S} (C) = ζ_{A} (J) > ⌊ A ⌉ (n - t)

. Then

⌊\frac{d_{L S} (C) - 1}{⌊ A ⌉}⌋ \geq n - t

. By Theorem 4, we have C is MDS. □

Example 7.

Let

A = 2^{[3]} 3^{[1]} 5^{[1]}

with

⌊ A ⌉ = 5

. Consider

C_{1} = {(0, 0, x, y, z, w) | x, y, z, w \in Z_{120}}

, and

C_{2} = {(0, 0, 0, x, y, z) | x, y, z \in Z_{120}}

Under the poset $P_{2} \times P_{A}$ , where $P_{2} = ([6], ⪯_{P_{2}})$ as in Figure 1, it is clear that $d_{L S} (C_{1}) = 11$ and $d_{L S} (C_{2}) = 16$ . Then $C_{1}$ and $C_{2}$ are MDS. Moreover, the poset $P_{2} \times P_{A}$ has exactly one ideal $I = 〈 {(1, A), (2, A)} 〉 〉$ such that $ζ_{A} (I) = 10$ , and has exactly one ideal $J = 〈 (3, A) 〉$ such that $ζ_{A} (J) = 15$ . Clearly, $C_{1}$ and $C_{2}$ are I-perfect and J-perfect, respectively.
Under the poset $P_{3} \times P_{A}$ , where $P_{3} = ([6], ⪯_{P_{3}})$ as in Figure 1, it is clear that $d_{L S} (C_{1}) = 1$ and $d_{L S} (C_{2}) = 6$ . Then $C_{1}$ and $C_{2}$ are not MDS. In addition, there are two ideals $I_{1} = 〈 (6, A) 〉$ and $I_{2} = 〈 (4, A) 〉$ such that $ζ_{A} (I_{1}) = 10$ and $ζ_{A} (I_{2}) = 15$ in which $C_{1}$ is not $I_{1}$ -perfect, and $C_{2}$ is not $I_{2}$ -perfect.

Observe that with the full count property of ideals in

P \times P_{A}

, it was a main tool to study the MDS

L S

-poset code

C \subseteq Z_{m}^{n}

where

| C | = m^{t}

0 \leq t \leq n

Next, we denote by

C_{n}

the chain poset

P = ([n], ⪯_{P})

with

min P = 1

. Observe that every ideal I with full count in

C_{n} \times P_{A}

has a unique maximal element. Suppose

r = t ⌊ A ⌉ \leq n ⌊ A ⌉

for some

t \in N

. There is only one ideal I with full count such that

ζ_{A} (I) = t ⌊ A ⌉

. It follows that

B_{r, L S} (u) = B_{I, L S} (u)

for all

u \in Z_{m}^{n}

The following results are some immediate consequences.

Proposition 6.

In the space

Z_{m}^{n}

with the poset

C_{n} \times P_{A}

, given an ideal I with full count in

C_{n} \times P_{A}

, let C be an I-perfect

L S

-poset code of

Z_{m}^{n}

. Then

1.: C is an $ζ_{A} (I)$ -perfect $L S$ -poset code.
2.: C is an MDS $L S$ -poset code.

Recall that the cardinality of an r-ball with center

u \in Z_{m}^{n}

as in (2). By considering m as a prime power, we obtain the following result.

Proposition 7.

In the space

Z_{q^{β}}^{n}

with the poset

C_{n} \times P_{A}

, if

β > 1

, the cardinality of

B_{r, L S} (0)

q^{r - t}

, where

(t - 1) (β + 1) < r \leq t (β + 1)

for

0 < t \leq n

Proof.

A = q^{[β]}

, we have the poset

(P (A), \subseteq_{A})

becomes a chain. This implies that every ideal I in

C_{n} \times P_{A}

contains a unique maximal element. Since

β > 1

⌊ A ⌉ = β + 1

. We write

r = (t - 1) (β + 1) + s + 1

, where

0 \leq s \leq β

. From (2), we have

\begin{matrix} | B_{r, L S} (0) | & = 1 + \sum_{i = 1}^{r} | A_{I^{i} (1)} | \\ = 1 + \sum_{j = 0}^{t - 2} q^{j β} [ϕ (q) + ϕ (q^{2}) + \dots + ϕ (q^{β})] + q^{(t - 1) β} [ϕ (q) + ϕ (q^{2}) + \dots + ϕ (q^{s})] \\ = 1 + \sum_{j = 0}^{t - 2} q^{j β} [q^{β} - 1] + q^{(t - 1) β} [q^{s} - 1] \\ = 1 + [q^{β} - 1] (\frac{q^{(t - 1) β} - 1}{q^{β} - 1}) + q^{(t - 1) β} [q^{s} - 1] = q^{r - t} . \end{matrix}

□

From above Proposition, although

| B_{r, L S} (0) |

divides

q^{β n}

, by Corollary 1, there is no r-perfect

L S

-poset code of

Z_{q^{β}}^{n}

β + 1

does not divide r.

Next, we have thus established the sufficient condition of ideals with partial count in

C_{n} \times P_{A}

for an

L S

-poset code of

Z_{m}^{n}

to be an r-perfect.

Theorem 6.

For

m = p_{1}^{β_{1}} p_{2}^{β_{2}} \dots p_{k}^{β_{k}}

with

k > 1

, in the space

Z_{m}^{n}

with the poset

C_{n} \times P_{A}

, if there is a unique

p_{i}

such that

β_{i} = 1

, then there is an

(t ⌊ A ⌉ + 1)

-perfect

L S

-poset code of

Z_{m}^{n}

for

0 \leq t < n

Proof.

By the assumption, we have

⌊ A ⌉ = | A |

, and

p_{i}^{[1]}

is the unique element of

P (A)

such that

⌊ p_{i}^{[1]} ⌉ = 1

. Let

0 \leq t < n

. By applying Proposition 2 (2.1), there is a unique ideal I with partial count in

C_{n} \times P_{A}

such that

ζ_{A} (I) = t ⌊ A ⌉ + 1

. Since

p_{i}^{[1]} \in E_{A}

, by Theorem 3, it follows that

B_{I, L S}^{⊥} (0)

becomes an

(t ⌊ A ⌉ + 1)

-perfect

L S

-poset code. □

Example 8.

In the space

Z_{12}^{3}

with the poset

C_{3} \times P_{A}

and the mset

A = 2^{[2]} 3^{[1]}

, we consider

C = {000, 003, 006, 009}

. Observe that

d_{L S} (C) = w_{L S} (006) = 8

. Then we have

B_{7, L S} (000) = Z_{12} \times Z_{12} \times 〈 4 〉

B_{7, L S} (003) = Z_{12} \times Z_{12} \times (3 + 〈 4 〉)

B_{7, L S} (006) = Z_{12} \times Z_{12} \times (6 + 〈 4 〉)

, and

B_{7, L S} (009) = Z_{12} \times Z_{12} \times (9 + 〈 4 〉)

. It can see that C is an 7-perfect

L S

-poset code. Moreover, C is also an I-perfect

L S

-poset code when the ideal

I = 〈 (3, 3^{[1]}) 〉

with

ζ_{A} (I) = 7

In the space

Z_{60}^{3}

with the poset

C_{3} \times P_{A}

and the mset

A = 2^{[2]} 3^{[1]} 5^{[1]}

, let

D = {(0, 0, 0), (0, 0, 15), (0, 0, 30), (0, 0, 45)}

. By Theorem 3, it can see that D is an J-perfect

L S

-poset code, where the ideal

J = 〈 (3, 3^{[1]} 5^{[1]}) 〉

such that

ζ_{A} (J) = 10

. Observe that

d_{L S} (D) = w_{L S} (0, 0, 30) = 10

. However, the space

Z_{60}^{3}

is not covered by the union of r-balls centered at the codewords of D for any

r < 10

Funding

This research received no external funding.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Conflicts of Interest

The authors declare no conflicts of interest.

References

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D’Oliveira, R.G.L.; Firer, M. The packing radius of a code and partitioning problems: the case for poset metrics on finite vector spaces. Discrete Math. 2015, 338, 2143–2167. [CrossRef]
Hyun, J.Y.; Kim, H.K. Maximum distance separable poset codes. Des. Codes Cryptogr. 2008, 48(3), 247–261. [CrossRef]
Kim, H.K.; Krotov, D.S. The poset metrics that allow binary codes of codimension m to be m-, (m − 1)-, or (m − 2)-perfect. IEEE Trans. Inf. Theory 2008, 54(11), 5241–5246.
Panek, L.; Firer, M.; Kim, H.K.; and Hyun, J.Y. Groups of linear isometries on poset structures. Discrete Math. 2008, 308(18), 4116–4123. [CrossRef]
Panek, L.; Pinheiro, J.A. General approach to poset and additive metrics. IEEE Trans. Inf. Theory 2020, 66(11), 6823–6834. [CrossRef]
Sudha, I.G.; Selvaraj, R.S. Code with a pomset metric and constructions. Des. Codes Cryptogr. 2018, 86, 875–892. [CrossRef]
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Figure 1. The posets

P_{1}, P_{2}, P_{3}

Figure 1. The posets

P_{1}, P_{2}, P_{3}

Figure 2. Weight relationship tree

Figure 3. The lattice of nontrivial subgroups for

Z_{120}

Figure 3. The lattice of nontrivial subgroups for

Z_{120}

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Codes with Weighted Poset Metrics Based on the Lattice of Subgroups of Z_m

Abstract

Keywords:

Subject:

1. Introduction

1.1. Poset Metrics

1.2. Multisets and Pomsets

1.3. Pomset Metrics

2. An Ordinal Product

3. Supports and Weights

4. r-Balls and I-Balls

5. MDS L S -Poset Codes and Codes in Chain Poset Structure

Funding

Data Availability Statement

Conflicts of Interest

References

MDPI Initiatives

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5. MDS $L S$ -Poset Codes and Codes in Chain Poset Structure