Narrow Results

In this paper, we determine the complete weight enumerator of the space ${𝔽}_q^n$ endowed by the $(P,w)$-metric, also known as the $(P,w)$-space. We define the $r$-balls and determine their cardinality. The Singleton bound is established on codes in $(P,w)$-space equipped with an arbitrary poset $P$ and a necessary condition for a code to be maximum distance separable (MDS) is obtained. Further, we define $I$-balls where $I$ is an ideal in $P$ and established the relationship between $I$-perfect codes and MDS $(P,w)$-codes. The duality theorem with respect to $(P,w)$-metric is arrived at. Also, the case when $P$ is a chain is explored leading to results on MDS codes and perfect codes in Niederreiter–Rosenbloom– Tsfasman (NRT) weighted space. Finally, weight distribution of MDS NRT codes is determined.

Let ${𝔽}_{q^l}$ be a finite field of cardinality $q^l$, where $q$ is a power of a prime and $l$ is a prime integer. An ${𝔽}_q$-linear code over ${𝔽}_{q^l}$ of length $n$ is an ${𝔽}_q$-subspace of ${𝔽}_{q^l}^n$. If an ${𝔽}_q$-linear code over ${𝔽}_{q^l}$ with parameters $(n,q^k,d_H)$ meets the Singleton bound, i.e., $d_H=n-\lceil \frac{k}{l}\rceil +1$, then it is called an MDS code. In this paper, by use of a Fourier matrix and a canonical form decomposition of constacyclic ${𝔽}_q$-linear codes over ${𝔽}_{q^l}$, we construct some optimal ${𝔽}_q$-linear codes over ${𝔽}_{q^l}$ which have the same parameters as the MDS codes over ${𝔽}_{q^l}$.

Let $p$ be an odd prime, $s$ and $m$ be positive integers and $\lambda$ be a nonzero element of ${𝔽}_{p^m}$. The $\lambda$-constacyclic codes of length $p^s$ over ${𝔽}_{p^m}$ are linearly ordered under set theoretic inclusion as ideals of the chain ring ${𝔽}_{p^m}[x]/〈x^{p^s}-\lambda 〉$. Using this structure, the symbol-triple distances of all such $\lambda$-constacyclic codes are established in this paper. All maximum distance separable symbol-triple constacyclic codes of length $p^s$ are also determined as an application.

Using the Calderbank–Shor–Steane (CSS) construction, pure q-ary asymmetric quantum error-correcting codes attaining the quantum Singleton bound are constructed. Such codes are called pure CSS asymmetric quantum maximum distance separable (AQMDS) codes. Assuming the validity of the classical maximum distance separable (MDS) Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for.

Negacyclic codes of length $2^s$ over the Galois ring $\mathrm{GR}(2^a,m)$ are linearly ordered under set-theoretic inclusion, i.e., they are the ideals $⟨{(x+1)}^i⟩$, $0\le i\le 2^{s}a$, of the chain ring $\mathrm{GR}(2^a,m)[x]/⟨x^{2^s}+1⟩$. This structure is used to obtain the symbol-pair distances of all such negacyclic codes. Among others, for the special case when the alphabet is the finite field ${\mathbb{F}}_{2^m}$ (i.e., $a=1$), the symbol-pair distance distribution of constacyclic codes over ${\mathbb{F}}_{2^m}$ verifies the Singleton bound for such symbol-pair codes, and provides all maximum distance separable symbol-pair constacyclic codes of length $2^s$ over ${\mathbb{F}}_{2^m}$.

Recently, several authors used linear codes to construct secret sharing schemes. It is known that if each nonzero codeword of a code $C$ is minimal, then the dual code $C^{\perp }$ is suitable for secret sharing. To seek such codes Ashikhmin–Barg give a sufficient condition from weights; in ${[n,k,d]}_q$ code $C$, let $w_{min}$ and $w_{max}$ be the minimum and maximum nonzero weights, respectively. If

Given a regular multiset $M$ on $[n]=\{1,2,\dots ,n\}$, a partial order $R$ on $M$, and a label map $\pi :[n]\to \mathbb{N}$ defined by $\pi (i)=k_i$ with ${\sum }_{i=1}^n\pi (i)=N$, we define a pomset block metric $d_{(Pm,\pi )}$ on the direct sum ${\mathbb{Z}}_m^{k_1}\oplus {\mathbb{Z}}_m^{k_2}\oplus \cdots \oplus {\mathbb{Z}}_m^{k_n}$ of ${\mathbb{Z}}_m^N$ based on the pomset $\mathbb{P}=(M,R)$. The pomset block metric extends the classical pomset metric introduced by Sudha and Selvaraj and generalizes the poset block metric introduced by Alves et al. over ${\mathbb{Z}}_m$. The space $({\mathbb{Z}}_m^N,d_{(Pm,\pi )})$ is called the pomset block space and we determine the complete weight distribution of it. Further, $I$-perfect pomset block codes for ideals with partial and full counts are described. Then, for block codes with chain pomset, the packing radius and Singleton bound are established. The relation between maximum distance separable (MDS) codes and $I$-perfect codes for any ideal $I$ is investigated. Moreover, the duality theorem for an MDS pomset block code is established when all the blocks have the same size.

Let ${𝔽}_{p^m}$ denote a finite field, where $p$ is a prime and $m$ is a positive integer. In this paper, we study idempotent decomposition of the finite, commutative, non-local ring, ${ℛ}={𝔽}_{p^m}[u,v]/〈u^2-\delta u,v^2-\gamma v,uv-vu〉$, where $\delta ,\gamma \in {𝔽}_{p^m}^{\ast }$. This paper consists of the structure of $\lambda$-constacyclic (CC) codes over ${ℛ}$, where $\lambda ={\xi }_1{\lambda }_1+{\xi }_2{\lambda }_2+{\xi }_3{\lambda }_3+{\xi }_4{\lambda }_4$. The structure of dual of CC codes is used to determine the Euclidean hull of such codes. Then, the Hamming distance and symbol-pair distance for the $\lambda$-CC codes of length $p^s$ over ${ℛ}$ is determined. Additionally, conditions for repeated root $\lambda$-CC code to be maximal distance separable (MDS) in terms of Hamming distance and symbol-pair distance are provided. Finally, we present examples of several linear and symbol-pair codes some of which are MDS with respect to Hamming and symbol-pair distance.

In this paper, we establish the Singleton bound for pomset block codes ($(Pm,\pi )$-codes) of length $N$ over the ring ${\mathbb{Z}}_m$. An upper and a lower bound on the minimum distance of $(Pm,\pi )$-Maximum distance separable (MDS) codes are derived. Moreover, we prove that an MDS $(Pm,\pi )$-code is necessarily an MDS $(P,\pi )$-code. We also investigate the relationship between the $I$-perfect codes and $t$-perfect codes. Given an ideal with partial count and full count, we investigate how MDS and $I$-perfect codes relate to one another. Duality theorem is derived for an MDS $(Pm,\pi )$-code when all the blocks are of same length, and the distribution of codewords among $I$-balls is analyzed as well. Finally, for a chain pomset, we study the maximum distance separability, packing radius and minimum distance of pomset block codes.

A $q$-ary linear $[n,k,n-k-(i-1)]$ code ${ℒ}$ is known as $A^i\mathrm{\text{MDS}}$ code. If $d=n-k+1,$ the code is called a maximum distance separable code. $\mathrm{\text{MDS}}$ codes are known to be useful in the secret-sharing schemes and coding theory. In this paper, we classify all binary self-orthogonal $A^3\mathrm{\text{MDS}}$ codes. We have introduced a subclass of $A^3\mathrm{\text{MDS}}$ codes, called $N^3\mathrm{\text{STMDS}}$ codes, to form a special type of secret-sharing scheme where for any participant, there exist $(k-1)$ other participants who can collectively reveal the secret by combining their shares and there exists another set of $(k+2)$ participants including the given participant, who cannot reveal the secret.