On the Hamming distances of repeated-root constacyclic codes of length 4ps

Let $p$ be an odd prime, $s$, $m$ be positive integers, $\gamma ,\lambda$ be nonzero elements of the finite field ${\mathbb{F}}_{p^m}$ such that ${\gamma }^{p^s}=\lambda$. In this paper, we show that, for any positive integer $\eta$, the Hamming distances of all repeated-root $\lambda$-constacyclic codes of length $\eta p^s$ can be determined by those of certain simple-root $\gamma$-constacyclic codes of length $\eta$. Using this result, Hamming distances of all constacyclic codes of length $4p^s$ are obtained. As an application, we identify all MDS $\lambda$-constacyclic codes of length $4p^s$.     

On the symbol-pair distances of repeated-root constacyclic codes of length 2ps

Let $p$ be an odd prime, and $\lambda$ be a nonzero element of the finite field ${\mathbb{F}}_{p^m}$. The $\lambda$-constacyclic codes of length $2p^s$ over ${\mathbb{F}}_{p^m}$ are classified as the ideals of quotient ring $\frac{{\mathbb{F}}_{p^m}\left[x\right]}{〈x^{2p^s}?\lambda 〉}$ in terms of their generator polynomials. Based on these generator polynomials, the symbol-pair distances of all such $\lambda$-constacyclic codes of length $2p^s$ are obtained in this paper. As an application, all MDS symbol-pair constacyclic codes of length $2p^s$ over ${\mathbb{F}}_{p^m}$ are established, which produce many new MDS symbol-pair codes with good parameters.     

A class of optimal linear codes of length one above the Griesmer bound

In this paper, we determine the smallest lengths of linear codes with some minimum distances. We construct a [g _q (k, d) + 1, k, d] _q code for sq ^k-1 − sq ^k-2 − q ^s  − q ² + 1 ≤ d ≤ sq ^k-1 − sq ^k-2 − q ^s with 3 ≤ s ≤ k − 2 and q ≥ s + 1. Then we get n _q (k, d) = g _q (k, d) + 1 for (k − 2)q ^k-1 − (k − 1)q ^k-2 − q ² + 1 ≤ d ≤ (k − 2)q ^k-1 − (k − 1)q ^k-2, k ≥ 6, q ≥ 2k − 3; and sq ^k-1 − sq ^k-2 − q ^s  − q + 1 ≤ d ≤ sq ^k-1 − sq ^k-2 − q ^s , s ≥ 2, k ≥ 2s + 1 and q ≥ 2s − 1. This work was partially supported by the Com²MaC-SRC/ERC program of MOST/KOSEF (grant ^# R11-1999-054) and was partially supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD)(KRF-2005-214-C00175).     

Non-invertible-element constacyclic codes over finite PIRs

In this paper we introduce the notion of λ-constacyclic codes over finite rings R for arbitrary element λ of R. We study the non-invertible-element constacyclic codes (NIE-constacyclic codes) over finite principal ideal rings (PIRs). We determine the algebraic structures of all NIE-constacyclic codes over finite chain rings, give the unique form of the sets of the defining polynomials and obtain their minimum Hamming distances. A general form of the duals of NIE-constacyclic codes over finite chain rings is also provided. In particular, we give a necessary and sufficient condition for the dual of an NIE-constacyclic code to be an NIE-constacyclic code. Using the Chinese Remainder Theorem, we study the NIE-constacyclic codes over finite PIRs. Furthermore, we construct some optimal NIE-constacyclic codes over finite PIRs in the sense that they achieve the maximum possible minimum Hamming distances for some given lengths and cardinalities.     

常重复合码的构造和界

摘要给出了一种Chebyshev距离下的常重复合码的构造,并在其基础上讨论了它的译码算法和优化处理.考虑了Chebyshev距离下的界及其改进.研究了具有Chebyshev距离和Hamming距离的常重复合码的构造,给出了Hamming距离为4的常重复合码的一个结论.     

Affine blocking sets, three-dimensional codes and the Griesmer bound

We construct families of three-dimensional linear codes that attain the Griesmer bound and give a non-explicit construction of linear codes that are one away from the Griesmer bound. All these codes contain the all-1 codeword and are constructed from small multiple blocking sets in AG(2,q).     

On the minimum length of some linear codes

We determine the minimum length n _q (k, d) for some linear codes with k ≥ 5 and q ≥ 3. We prove that n _q (k, d) = g _q (k, d) + 1 for when k is odd, for when k is even, and for . This work was supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD). (KRF-2005-214-C00175). This research has been partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science under Contract Number 17540129.     

A weighted version of a result of Hamada on minihypers and on linear codes meeting the Griesmer bound

Minihypers were introduced by Hamada to investigate linear codes meeting the Griesmer bound. Hamada (Bull Osaka Women’s Univ 24:1–47, 1985; Discrete Math 116:229–268, 1993) characterized the non-weighted minihypers having parameters , with k−1 > λ₁ > λ₂ > ... > λ _h ≥ 0, as the union of a λ₁-dimensional space, λ₂-dimensional space, ..., λ _h -dimensional space, which all are pairwise disjoint. We present in this article a weighted version of this result. We prove that a weighted -minihyper , with k−1 > λ₁ > λ₂ > ... > λ _h ≥ 0, is a sum of a λ₁-dimensional space, λ₂-dimensional space, ..., and λ _h -dimensional space. This research was supported by the Project Combined algorithmic and theoretical study of combinatorial structures between the Fund for Scientific Research Flanders-Belgium (FWO-Flanders) and the Bulgarian Academy of Sciences. This research is also part of the FWO-Flanders project nr. G.0317.06 Linear codes and cryptography.     

Nonexistence of some linear codes over the field of order four

We consider the problem of determining $n_4(5,d)$, the smallest possible length $n$ for which an ${[n,5,d]}_4$ code of minimum distance $d$ over the field of order 4 exists. We prove the nonexistence of ${[g_4(5,d)+1,5,d]}_4$ codes for $d=31,47,48,59,60,61,62$ and the nonexistence of a ${[g_4(5,d),5,d]}_4$ code for $d=138$ using the geometric method through projective geometries, where $g_q(k,d)={\sum }_{i=0}^{k?1}?d∕q^i?$. This yields to determine the exact values of $n_4(5,d)$ for these values of $d$. We also give the updated table for $n_4(5,d)$ for all $d$ except some known cases.