Optimal Linear Codes and Their Hulls

This paper investigates the structures and properties of one-Lee weight codes and two-Lee weight projective codes over ℤ4. The authors first give the Pless identities on the Lee weight of linear codes over ℤ4. Then the authors study the necessary conditions for linear codes to have one-Lee weight and two-Lee projective weight respectively, the construction methods of one-Lee weight and two-Lee weight projective codes over ℤ4 are also given. Finally, the authors recall the weight-preserving Gray map from (ℤ 4 , Lee weight) to ( $\mathbb{F}_2^{2n} $ , Hamming weight), and produce a family of binary optimal oneweight linear codes and a family of optimal binary two-weight projective linear codes, which reach the Plotkin bound and the Griesmer bound.

Hermitian linear complementary dual codes are linear codes whose intersections with their Hermitian dual codes are trivial. The largest minimum weight among quaternary Hermitian linear complementary dual codes of dimension $2$ is known for each length. We give the complete classification of optimal quaternary Hermitian linear complementary dual codes of dimension $2$. Hermitian linear complementary dual codes are linear codes whose intersections with their Hermitian dual codes are trivial. The largest minimum weight among quaternary Hermitian linear complementary dual codes of dimension $2$ is known for each length. We give the complete classification of optimal quaternary Hermitian linear complementary dual codes of dimension $2$.

A linear code is optimal if it has the highest minimum distance of any linear code with a given length and dimension. We construct infinite families of optimal binary linear codes C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δc</sub> constructed from simplicial complexes in F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , where Δ is a simplicial complex in F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> and Δ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sup> the complement of Δ. We first find an explicit computable criterion for C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δc</sub> to be optimal; this criterion is given in terms of the 2-adic valuation of Σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j=1</sub> 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|Ai|-1</sup> , where the At's are maximal elements of Δ. Furthermore, we obtain much simpler criteria under various specific conditions on the maximal elements of Δ. In particular, we find that C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δc</sub> is a Griesmer code if and only if the maximal elements of Δ are pairwise disjoint and their sizes are all distinct. Specially, when f has exactly two maximal elements, we explicitly determine the weight distribution of C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δc</sub> .We present many optimal linear codes constructed by our method, and we emphasize that we obtain at least 32 new optimal linear codes.

This paper discusses optimal binary codes and pure binary quantum codes created using Steane construction. First, a local search algorithm for a special subclass of quasi-cyclic codes is proposed, then five binary quasi-cyclic codes are built. Second, three classical construction methods are generalized for new codes from old such that they are suitable for constructing binary self-orthogonal codes, and 62 binary codes and six subcode chains of obtained self-orthogonal codes are designed. Third, six pure binary quantum codes are constructed from the code pairs obtained through Steane construction. There are 66 good binary codes that include 12 optimal linear codes, 45 known optimal linear codes, and nine known optimal self-orthogonal codes. The six pure binary quantum codes all achieve the performance of their additive counterparts constructed by quaternary construction and thus are known optimal codes.

We consider locally repairable codes over small fields and propose constructions of optimal cyclic and linear codes in terms of the dimension for a given distance and length. Four new constructions of optimal linear codes over small fields with locality properties are developed. The first two approaches give binary cyclic codes with locality two. While the first construction has availability one, the second binary code is characterized by multiple available repair sets based on a binary Simplex code. The third approach extends the first one to q-ary cyclic codes including (binary) extension fields, where the locality property is determined by the properties of a shortened first-order Reed-Muller code. Non-cyclic optimal binary linear codes with locality greater than two are obtained by the fourth construction.

&lt;p style='text-indent:20px;'&gt;Recently, some infinite families of optimal binary linear codes are constructed from simplicial complexes. Afterwards, the construction method was extended to using arbitrary posets. In this paper, based on a generic construction of linear codes, we obtain four classes of optimal binary linear codes by using the posets of two chains. Two of them induce Griesmer codes which are not equivalent to the linear codes constructed by Belov. Those codes are exploited to construct secret sharing schemes in cryptography as well.&lt;/p&gt;

Recently linear codes with locality properties have attracted a lot of interest due to their desirable applications in distributed storage systems. An [n, k, d] linear code with (r, δ)-locality can enable the local recovery of a failed node in case of more than one node failures. In this paper, we study the theoretical bounds and constructions of linear codes with (r, δ)-locality for all code symbols. A parity-check matrix approach is employed to present an alternate simple proof of the Singleton-like bound for linear codes with all symbol (r, δ)-locality. A refined Singleton-like bound is given for the case that r | k and r + δ − 1 ∤ n. Base on the new proof technique, we enumerate all the possible two classes of optimal binary linear codes meeting the Singleton-like bound. In other words, except the proposed two classes of optimal binary linear codes, there is no other binary linear codes with minimum distance d = n — k — ([k/r] — 1)(δ− 1) + 1.

Recently, some infinite families of minimal and optimal binary linear codes were constructed from simplicial complexes by Hyun et al. We extend this construction method to arbitrary posets. Especially, anti-chains are corresponded to simplicial complexes. In this paper, we present two constructions of binary linear codes from hierarchical posets of two levels. In particular, we determine the weight distributions of binary linear codes associated with hierarchical posets with two levels. Based on these results, we also obtain some optimal and minimal binary linear codes not satisfying the condition of Ashikhmin–Barg.

It is proved that there are no unknown perfect (Hamming-)error-correcting codes over finite fields.

New ternary codes of dimension 6 are presented which improve the bounds on optimal linear codes. These codes belong to the class of quasi-twisted (QT) codes, and have been constructed using a greedy algorithm. This work extends previous results on QT codes of dimension 6. In particular, several new two-weight QT codes are presented. Numerous new optimal codes which meet the Griesmer bound are given, as well as others which establish lower bounds on the maximum minimum distance.

Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called duadic double circulant codes, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual codes, optimal linear codes, and linear codes with the best known parameters in a systematic way. We describe a method to construct duadic double circulant codes using 4-cyclotomic cosets and give certain duadic double circulant codes over \(\mathbb{F}_{2}\), \(\mathbb{F}_{3}\), \(\mathbb{F}_{4}\), \(\mathbb{F}_{5}\), and \(\mathbb{F}_{7}\). In particular, we find a new ternary self-dual [76,38,18] code and easily rediscover optimal binary self-dual codes with parameters [66,33,12], [68,34,12], [86,43,16], and [88,44,16] as well as a formally self-dual binary [82,41,14] code.

Hulls of linear codes have been of interest and extensively studied due to their wide applications. In this paper, we focus on constructions and optimality of linear codes with hull dimension one over small finite fields. General constructions for such codes are given together with the analysis on their parameters. Optimal linear $$[n,2,d]_q$$ codes with hull dimension one are presented for all positive integers $$n\ge 3$$ and $$q\in \{2,3\}$$ . Moreover, for $$q=2$$ , the enumeration of such optimal codes is given up to equivalence.

Constacyclic codes are an important class of linear codes in coding theory. Many optimal linear codes are directly derived from constacyclic codes. In this paper, a new Gray map between codes over Fp + uFp + u2Fp and codes over Fp is defined, where p is an odd prime. By means of this map, it is shown that the Gray image of a linear (1+u+u2)-constacyclic code over Fp + uFp + u2Fp of length n is a repeated-root cyclic code over Fp of length pn. Furthermore, some examples of optimal linear cyclic codes over F3 from (1 + u + u2)-constacyclic codes over F3 + uF3 + u2F3 are given.

A construction of some [ n, k, d; q]-codes meeting the Griesmer bound

Optimal non-projective linear codes constructed from down-sets