We construct new linear codes with high minimum distance d. In at least 12 cases these codes improve the minimum distance of the previously known best linear codes for fixed parameters n,k. Among these new codes there is an optimal ternary [88,8,54] code.
We develop an algorithm, which starts with already good codes C, i.e. codes with high minimum distance d for given length n and dimension k over the field GF(q). The algorithm is based on the newly defined (l,s)-extension. This is a generalization of the well-known method of adding a parity bit in the case of a binary linear code of odd minimum weight. (l,s)-extension tries to extend the generator matrix of C by adding l columns with the property that at least s of the l letters added to each of the codewords of minimum weight in C are different from 0. If one finds such columns the minimum distance of the extended code is d+s provided that the second smallest weight in C was at least d+s. The question whether such columns exist can be settled using a Diophantine system of equations.
Entanglement-assisted quantum error correcting codes (EAQECCs) constructed from Reed-Solomon codes and BCH codes are considered in this work. It is provided a complete and explicit formula for the parameters of EAQECCs coming from any Reed-Solomon code, for the Hermitian metric, and from any BCH code with extension degree $2$ and consecutive cyclotomic cosets, for both the Euclidean and the Hermitian metric. The main task in this work is the computation of a completely general formula for $c$, the minimum number of required maximally entangled quantum states.
In this paper a framework to study the dual of skew cyclic codes is proposed. The transposed Hamming ring extensions are based in the existence of an anti-isomorphism of algebras between skew polynomial rings. Our construction is applied to left ideal convolutional codes, skew constacyclic codes and skew Reed-Solomon code, showing that the dual of these codes belong to the same class.
Protograph low-density-parity-check (LDPC) are considered to design near-capacity low-rate codes over the binary erasure channel (BEC) and binary additive white Gaussian noise (BIAWGN) channel. For protographs with degree-one variable nodes and doubly-generalized LDPC (DGLDPC) codes, conditions are derived to ensure equality of bit-error threshold and block-error threshold. Using this condition, low-rate codes with block-error threshold close to capacity are designed and shown to have better error rate performance than other existing codes.
This paper consists of three components. In the first, we give an adelic interpretation of the classical extension class associated to extension of locally free sheaves on curves. Then, in the second, we use this construction on adelic extension classes to write down explicitly adelic representors in $GL_r(A)$ for Atiyah bundles $I_r$ on elliptic curves. All these works make sense over any base fields. Finally, as an application, for $m\geq 1$, we construct the global sections of $I_r(mQ)$ in local terms and apply it to obtain rank $r$ MDS codes based on the codes spaces $C_{F;r}(D; I_r(mQ))$ …
Let $p$ be a prime number and $s> 0$ an integer. In this short note, we investigate one-point geometric Goppa codes associated with an elementary abelian $p$-extension of $\mathbb{F}_{p^{s}}(x)$. We determine their dimension and the exact minimum distance in a few cases. These codes are a special case of weak Castle codes. We also list the exact values of the second generalized Hamming weight of these codes in a few cases. Simple criteria for the self-duality and the quasi-self-duality of these codes are also provided. Furthermore, we construct examples of quantum codes, convolutional codes, a…
对于由$y^m = f (x)$定义的Kummer扩张，其中$f (x)$是有限域$\mathbb{F}_q$上的可分多项式，我们计算了两个完全分歧处的Weierstrass间隙数.对于许多完全分歧的地方，我们给出了一个标准，找到纯差距在这些点和目前的家庭纯差距。然后，我们应用我们的结果来构造具有良好参数的多点代数几何码。
Recently, Skabelund defined new maximal curves which are cyclic extensions of the Suzuki and Ree curves. Previously, the now well-known GK curves were found as cyclic extensions of the Hermitian curve. In this paper, we consider locally recoverable codes constructed from these new curves, complementing that done for the GK curve. Locally recoverable codes allow for the recovery of a single symbol by accessing only a few others which form what is known as a recovery set. If every symbol has at least two disjoint recovery sets, the code is said to have availability. Three constructions are descr…
We consider the leakage resilience of AG code-based ramp secret sharing schemes extending the leakage resilience of linear threshold secret sharing schemes over prime fields done by Benhamouda et al. Since there is not any explicit efficient construction of AG codes over prime fields, we consider constructions over prime fields with the help of concatenation method and those over field extensions. Extending the Fourier analysis done by Benhamouda et al., concatenated algebraic geometric codes over prime fields do produce some nice leakage-resilient secret sharing schemes. One natural and curio…
This paper extends the study of rank-metric codes in extension fields $\mathbb{L}$ equipped with an arbitrary Galois group $G = \mathrm{Gal}(\mathbb{L}/\mathbb{K})$. We propose a framework for studying these codes as subspaces of the group algebra $\mathbb{L}[G]$, and we relate this point of view with usual notions of rank-metric codes in $\mathbb{L}^N$ or in $\mathbb{K}^{N\times N}$, where $N = [\mathbb{L} : \mathbb{K}]$. We then adapt the notion of error-correcting pairs to this context, in order to provide a non-trivial decoding algorithm for these codes. We then focus on the case where $G$…
In this paper, linear index codes with multiple senders are studied, where every receiver receives encoded messages from all senders. A new fitting matrix for the multiple senders is proposed and it is proved that the minimum rank of the proposed fitting matrices is the optimal codelength of linear index codes for the multiple senders. In addition, a new type of a side information graph related with the optimal codelength is proposed and whether given side information is critical or not is studied. Furthermore, linear index codes for the cellular network scenario are studied, where each receiv…
The two-sender groupcast index coding problem (TGICP) consists of a set of receivers, where all the messages demanded by the set of receivers are distributed among the two senders. The senders can possibly have a set of messages in common. Each message can be demanded by more than one receiver. Each receiver has a subset of messages (known as its side information) and demands a message it does not have. The objective is to design scalar linear codes at the senders with the minimum aggregate code length such that all the receivers are able to decode their demands, by leveraging the knowledge of…
Linear codes over finite extension fields have widespread applications in theory and practice. In some scenarios, the decoder has a sequential access to the codeword symbols, giving rise to a hierarchical erasure structure. In this paper we develop a mathematical framework for hierarchical erasures over extension fields, provide several bounds and constructions, and discuss potential applications in distributed storage and flash memories. Our results show intimate connection to Universally Decodable Matrices, as well as to Reed-Solomon and Gabidulin codes.
Suppose that we have two training sequences generated by parametrized distributions $P_{θ^*}$ and $P_{ξ^*}$, where $θ^*$ and $ξ^*$ are unknown true parameters. Given training sequences, we study the problem of classifying whether a test sequence was generated according to $P_{θ^*}$ or $P_{ξ^*}$. This problem can be thought of as a hypothesis testing problem and our aim is to analyze the weighted sum of type-I and type-II error probabilities. Utilizing the analysis of the codeword lengths of the Bayes code, our previous study derived more refined bounds on the error probability than known previ…
