Note on Niederreiter-Xing's Propagation Rule for Linear Codes

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 $2$.

In this work we continue the study of a new class of codes, called codes over graphs. Here we consider storage systems where the information is stored on the edges of a complete directed graph with n nodes. The failure model we consider is of node failures which are erasures of all edges, both incoming and outgoing, connected to the failed node. It is said that a code over graphs is a ρ-node-erasure-correcting code if it can correct the failure of any ρ nodes in the graphs of the code. While the construction of such optimal codes is an easy task if the field size is O(n2), our main goal in the paper is the construction of codes over smaller fields. In particular, our main result is the construction of optimal binary codes over graphs which correct two node failures with a prime number of nodes.

Nonlinear channel codes are employed to protect two information fields that are related to the daylight saving time (DST) and leap second notifications within the enhanced WWVB broadcast format. The general codebook design criteria and the construction of these codes using integer linear program (ILP) is formulated. Specifically, two different variations are presented: i) minimizing the union bound of probability of word error (PWE), which is a tighter bound but has high computational complexity and ii) minimizing an analytical closed form approximation to the PWE, which has much lower complexity but is a looser bound on the exact PWE. By leveraging the highly non-uniform a-priori probabilities of the different messages being transmitted in these two fields, the proposed nonlinear codes provide coding gains of about 2–3 dB in the presence of additive white Gaussian noise (AWGN). When compared to checksum codes with the same rate, the nonlinear codes offer performance advantages of about 1–2 dB for the DST notification field. In the presence of impulsive noise, simulations and analysis show that the probability of word error when employing the proposed nonlinear code is lower by many orders of magnitude compared with that of the uncoded system.

An index coding problem is symmetric if relative to its position, each receiver has identical sets of wanted messages and side information. We provide an algorithm to construct an optimal length scalar linear code for single unicast symmetric index coding problems with neighboring interference. First we compute the minimum number of independent dimensions required to avoid interference and show this number to be equal to the minimum possible length of a scalar linear index code. A code construction is provided that gives the optimal scalar linear index code. Further, we have identified the one-sided neighboring antidotes single unicast symmetric index coding problem as a special case of the neighboring interference problem and have constructed an optimal scalar linear index code for the former.

In this paper, we consider a communication system where a sender sends messages over a memoryless Gaussian point-to-point channel to a receiver and receives the output feedback over another Gaussian channel with known variance and unit delay. The sender sequentially transmits the message over multiple times till a certain error performance is achieved. The aim of our work is to design a transmission strategy to process every transmission with the information that was received in the previous feedback and send a signal so that the estimation error drops as quickly as possible. The optimal code is unknown for channels with noisy output feedback when the block length is finite. Even within the family of linear codes, optimal codes are unknown in general. Bridging this gap, we propose a family of linear sequential codes and provide a dynamic programming (DP) algorithm to solve for a closed form expression for the optimal code within a class of sequential linear codes. The optimal code discovered via DP is a generalized version of which the Schalkwijk-Kailath (SK) scheme is one special case with noiseless feedback; our proposed code coincides with the celebrated SK scheme for noiseless feedback settings.

Technology now-a-days has taken a rapid progress in several areas of wireless communications. Modeling of wireless channels plays a key role in the design of communication systems. This paper gives the analysis of certain coding methods used for determining the reliability of wireless channels. Analysis of various Linear Block codes involved in channel coding is determined and an optimal block Code is implemented with different modulation techniques with data transmitted over various channels. The Bit Error Rate (BER) which is calculated for several block codes for a specific (N, K) values is found to be very low for RS code for high Signal to Noise ratio (SNR). By varying the values of `N' and `K' for RS code, an optimal code in terms of Bit Error Rate (BER) versus Signal to Noise Ratio (SNR) is found. The results showed that RS (255,223) is the optimal code. The performance for this optimal code in the presence of few fading channels using BPSK and QPSK modulation techniques is also determined by modeling the codec.

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.

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

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.

In this letter, a method of constructing Hermitian dual containing quaternary linear codes from general quaternary linear codes is proposed. Based on this result, a construction method of binary linear quantum codes is obtained. Using three cyclic codes and one linear code over $\mathbb {F}_{4}$ that are not Hermitian dual containing, we construct four record breaking binary linear quantum codes. Furthermore, ten new record breaking binary additive quantum codes are derived from these four linear quantum codes by propagation rule.

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

The design of space-time codes for frequency flat, spatially correlated MIMO fading channels is considered. The focus of the paper is on the class of space-time block codes known as linear dispersion (LD) codes, introduced by Hassibi and Hochwald. The LD codes are optimized with respect to the mutual information between the inputs to the space-time encoder and the output of the channel. The use of the mutual information as both a design criterion and a performance measure is justified by allowing soft decisions at the output of the space-time decoder. A spatial Fourier (virtual) representation of the channel is exploited to allow for the analysis of MIMO channels with quite general fading statistics. Conditions, known as generalized orthogonal conditions (GOC's), are derived for an LD code to achieve an upper bound on the mutual information, with the understanding that LD codes that achieve the upper bound, if they exist, are optimal. Explicit code constructions and properties of the optimal power allocation schemes are also derived. In particular, it is shown that optimal LD codes correspond to beamforming to a single virtual transmit angle at low SNR, and a necessary and sufficient condition for beamforming to be optimal is provided. Due to the nature of the code construction, it is further observed that the optimal LD codes can be designed to adapt to the statistics of different scattering environments. Finally, numerical results are provided to illustrate the optimal code design for three examples of sparse scattering environments. The performance of the optimal LD codes for these scattering environments is compared with that of LD codes designed assuming the i.i.d. Rayleigh fading (rich scattering) model, and it is shown that the optimal LD codes perform significantly better. The optimal LD codes are also compared to beamforming LD codes and it is shown that beamforming is nearly optimal over a range of SNR's of interest.

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.

A class of optimal three-weight [q^k-1,k+1,q^{k-1}(q-1)-1] cyclic codes over {mathrm{I!F}}_q, with kge 2, achieving the Griesmer bound, was presented by Heng and Yue (IEEE Trans Inf Theory 62(8):4501–4513, 2016. https://doi.org/10.1109/TIT.2016.2550029). In this paper we study some of the subfield codes of this class of optimal cyclic codes when k=2. The weight distributions of the subfield codes are settled. It turns out that some of these codes are optimal and others have the best known parameters. The duals of the subfield codes are also investigated and found to be almost optimal with respect to the sphere-packing bound. In addition, the covering structure for the studied subfield codes is determined. Some of these codes are found to have the important property that any nonzero codeword is minimal, which is a desirable property that is useful in the design of a secret sharing scheme based on a linear code. Moreover, a specific example of a secret sharing scheme based on one of these subfield codes is given. Finally, a class of optimal two-weight linear codes over {mathrm{I!F}}_q, achieving the Griesmer bound, whose duals are almost optimal with respect to the sphere-packing bound is presented. Through a different approach, this class of optimal two-weight linear codes was reported very recently by Heng (IEEE Trans Inf Theory 69(2):978–994, 2023. https://doi.org/10.1109/TIT.2022.3203380). Furthermore, it is shown that these optimal codes can be used to construct strongly regular graphs.