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Linear codes achieving or nearly achieving the Singleton bound are interesting in both theory and practice. The objective of this paper is to construct several infinite families of MDS, near MDS and almost MDS codes from some special cyclic subgroups of ${\mathbb {F}}_{q^{2}}^{*}$. To this end, the augmentation and extension techniques are used. The codes in this paper have flexible parameters and their lengths could be large. The minimum linear locality of the codes constructed in this paper is also studied. Some infinite families of optimal linearly locally recoverable codes are obtained. Besides, some codes in this paper are proved to be proper for error detection. [ABSTRACT FROM AUTHOR]

In this paper, a family of $[{q}^{2}-1, 4, {q}^{2}-{q}-2]$ cyclic codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2}-1,{q}^{2}-5,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of cyclic codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. Some of the subfield codes are optimal and some have the best known parameters. It is shown that the subfield codes are equivalent to a family of primitive BCH codes and thus the parameters of the BCH codes are solved. The duals of the subfield codes are also optimal with respect to the sphere-packing bound. A family of $[{q}^{2}, 4, {q}^{2}-{q}-1]$ linear codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2},{q}^{2}-4,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of linear codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. Five infinite families of 2-designs are also constructed with three families of linear codes of this paper. [ABSTRACT FROM AUTHOR]

BCH codes are an important class of cyclic codes which have applications in satellite communications, DVDs, disk drives, and two-dimensional bar codes. Although BCH codes have been widely studied, their parameters are known for only a few special classes. Recently, Ding et al. made some new progress in BCH codes. However, we still have very limited knowledge on the dimension of BCH codes, not to mention the weight distribution of BCH codes. In this paper, we generalize the results on BCH codes from several previous papers. 1) The dimension of narrow-sense BCH codes of length $((q^{m}-1)/{\lambda })$ with designed distance $2\leq \delta \leq (({q^{\lceil (m+1)/2 \rceil }-1})/(\lambda)+1)$ is settled, where $\lambda $ is any factor of $(q-1)$. 2) The weight distributions of two classes of narrow-sense BCH codes of length $(({q^{m}-1})/2)$ with designed distance $\delta =(({(q-1)q^{m-1}-q^{\lfloor (m-1)/2\rfloor }-1})/2)$ and $\delta =(({(q-1)q^{m-1}-q^{\lfloor (m+1)/2\rfloor }-1})/2)$ are determined. 3) The weight distribution of a class of BCH codes of length $((q^{m}-1)/({q-1}))$ is determined. In particular, a subclass of this class of BCH codes is optimal with respect to the Griesmer bound. Some optimal linear codes obtained from this class of BCH codes are characterized. [ABSTRACT FROM AUTHOR]

The hull of a linear code is defined to be the intersection of the code and its dual, and was originally introduced to classify finite projective planes. The hull plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. It has been shown that these algorithms are very effective in general if the size of the hull is small. The objective of this paper is to present some sufficient and necessary conditions that linear codes and cyclic codes have one-dimensional hull. It is shown that there are no such binary or ternary cyclic codes. Based on these characterizations, some constructions of linear codes with one-dimensional hull were given by employing quadratic number fields, partial difference sets, and difference sets. We also construct cyclic codes with one-dimensional hull. Some optimal codes with one-dimensional hull are obtained. [ABSTRACT FROM AUTHOR]

Following the very recent studies on \mathbb Z2\mathbb Z4 -additive codes, \mathbb Z2\mathbb Z2[u] -linear codes have been introduced by Aydogdu et al. In this paper, we introduce and study the algebraic structure of cyclic, constacyclic codes and their duals over the R -module \mathbb {Z}_{2}^\alpha R^\beta where R=\mathbb {Z}_{2}+u\mathbb {Z}_{2}=\left \{{0,1,u,u+1}\right \} is the ring with four elements and u^2=0 . We determine the generating independent sets and the types and sizes of both such codes and their duals. Finally, we present a bound and an optimal family of codes attaining this bound and also give some illustrative examples of binary codes that have good parameters which are obtained from the cyclic codes in \mathbb Z_2^\alpha R^\beta . [ABSTRACT FROM AUTHOR]

Cyclic codes with a few weights are very useful in the design of frequency hopping sequences and the development of secret sharing schemes. In this paper, we mainly use Gauss sums to represent the Hamming weights of cyclic codes whose duals have two zeroes. A lower bound of the minimum Hamming distance is determined. In some cases, we give the Hamming weight distributions of the cyclic codes. In particular, we obtain a class of three-weight optimal cyclic codes achieving the Griesmer bound, which generalizes a Vega’s result, and several classes of cyclic codes with a few weights, which solve an open problem proposed by Vega. [ABSTRACT FROM AUTHOR]

In this paper, we extend two earlier works further in two directions and compute the weight distribution of these cyclic codes under more relaxed conditions. It is interesting to note that many cyclic codes in the family are optimal and have only a few non-zero weights. Besides using similar ideas, we carry out some subtle manipulation of certain exponential sums. [ABSTRACT FROM AUTHOR]

The generalized Hamming weights dr(C) of a linear code C are a natural generalization of the minimum Hamming distance d(C)[=d_{1}(C)] and have become an important research object in coding theory since Wei’s originary work in 1991. In this paper, two general formulas on dr(C) for irreducible cyclic codes are presented using Gauss sums and the weight hierarchy \{d_{1}(C), d_{2}(C), \ldots , d_{k}(C)\}~(k=\dim C)$ is completely determined for several cases. [ABSTRACT FROM AUTHOR]

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In this paper, let \(\Bbb F_r\) be a finite field with \(r=q^m\) . Suppose that \(g_1, g_2 \in \Bbb F_r^*\) are not conjugates over \(\Bbb F_q\) , \( \operatorname {ord}(g_1)=n_1\) , \( \operatorname {ord}(g_2)=n_2\) , \(d=\gcd (n_1, n_2)\) , and \(n={n_1n_2}/ d\) . Let \(\Bbb F_q(g_1)=\Bbb F_{q^{m_1}}, \Bbb F_q(g_2)=\Bbb F_{q^{m_2}}\) , and \(\text{T}_i\) denote the trace function from \(\Bbb F_{q^{m_i}}\) to \(\Bbb F_q\) for \(i=1, 2\) . We define a cyclic code \(\mathcal C_{(q, m, n_1, n_2)}=\{c(a,b) : a \in \Bbb F_{q^{m_1}}, b \in \Bbb F_{q^{m_2}}\},\) where \(c(a,b)=(\text {T}_1(ag_1^0)+\text {T}_2(bg_2^0), \text {T}_1(ag_1^1)+\text {T}_2(bg_2^1), \ldots , \text {T}_1(ag_1^{n-1})+\text {T}_2(bg_2^{n-1})).\) We mainly use Gauss periods to present the weight distribution of the cyclic code \(\mathcal C_{(q, m, n_1, n_2)}\) . As applications, we determine the weight distribution of cyclic code \(\mathcal C_{(q, m,q^{m_1}-1,q^{m_2}-1)}\) with \(\gcd (m_1,m_2)=1\) ; in particular, it is a three-weight cyclic code if \(\gcd (q-1,m_1-m_2)=1\) . We also explicitly determine the weight distributions of some classes of cyclic codes including several classes of four-weight cyclic codes. [ABSTRACT FROM AUTHOR]