The Weights in MDS Codes

<p style='text-indent:20px;'>The weight distribution of the cosets of maximum distance separable (MDS) codes is considered. In 1990, P.G. Bonneau proposed a relation to obtain the full weight distribution of a coset of an MDS code with minimum distance <inline-formula><tex-math id="M1">\begin{document}$ d $\end{document}</tex-math></inline-formula> using the known numbers of vectors of weights <inline-formula><tex-math id="M2">\begin{document}$ \le d-2 $\end{document}</tex-math></inline-formula> in this coset. In this paper, the Bonneau formula is transformed into a more structured and convenient form. The new version of the formula allows to consider effectively cosets of distinct weights <inline-formula><tex-math id="M3">\begin{document}$ W $\end{document}</tex-math></inline-formula>. (The weight <inline-formula><tex-math id="M4">\begin{document}$ W $\end{document}</tex-math></inline-formula> of a coset is the smallest Hamming weight of any vector in the coset.) For each of the considered <inline-formula><tex-math id="M5">\begin{document}$ W $\end{document}</tex-math></inline-formula> or regions of <inline-formula><tex-math id="M6">\begin{document}$ W $\end{document}</tex-math></inline-formula>, special relations more simple than the general ones are obtained. For the MDS code cosets of weight <inline-formula><tex-math id="M7">\begin{document}$ W = 1 $\end{document}</tex-math></inline-formula> and weight <inline-formula><tex-math id="M8">\begin{document}$ W = d-1 $\end{document}</tex-math></inline-formula> we obtain formulas of the weight distributions depending only on the code parameters. This proves that all the cosets of weight <inline-formula><tex-math id="M9">\begin{document}$ W = 1 $\end{document}</tex-math></inline-formula> (as well as <inline-formula><tex-math id="M10">\begin{document}$ W = d-1 $\end{document}</tex-math></inline-formula>) have the same weight distribution. The cosets of weight <inline-formula><tex-math id="M11">\begin{document}$ W = 2 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M12">\begin{document}$ W = d-2 $\end{document}</tex-math></inline-formula> may have different weight distributions; in this case, we proved that the distributions are symmetrical in some sense. The weight distributions of the cosets of MDS codes corresponding to arcs in the projective plane <inline-formula><tex-math id="M13">\begin{document}$ \mathrm{PG}(2,q) $\end{document}</tex-math></inline-formula> are also considered. For MDS codes of covering radius <inline-formula><tex-math id="M14">\begin{document}$ R = d-1 $\end{document}</tex-math></inline-formula> we obtain the number of the weight <inline-formula><tex-math id="M15">\begin{document}$ W = d-1 $\end{document}</tex-math></inline-formula> cosets and their weight distribution that gives rise to a certain classification of the so-called deep holes. We show that any MDS code of covering radius <inline-formula><tex-math id="M16">\begin{document}$ R = d-1 $\end{document}</tex-math></inline-formula> is an almost perfect multiple covering of the farthest-off points (deep holes); moreover, it corresponds to an optimal multiple saturating set in the projective space <inline-formula><tex-math id="M17">\begin{document}$ \mathrm{PG}(N,q) $\end{document}</tex-math></inline-formula>.</p>

Maximum distance separable (MDS) codes are those codes whose minimum distance attains the Singleton bound for a given length and given dimension. It is a well-known fact that the only binary linear block codes that are MDS are the simple parity codes and the repetition codes. Known MDS codes (the Reed-Solomon codes, for instance) are defined over larger alphabets. This approach requires that (i) the encoding and decoding procedures are performed as operations over a finite field and (ii) an update in a single information bit (e.g. in storage applications) requires an update in the redundancy symbols and usually affects a number of bits in each redundancy symbol. Thus, the optimal redundancy is achieved at the expense of additional complexity in the encoding/decoding procedures as well as in the number of redundancy bits affected by an update in the information. We construct MDS codes that have the following two properties: (i) the redundancy bits are computed by simple XOR operations; and (ii) an update in an information bit affects a minimal number of redundancy bits. >

In this paper, we propose two novel modulation concepts based on a simple maximum distance separable (MDS) code and show that these concepts can achieve better error performance than index modulation (IM) and related schemes. In the first concept, we use amplitude and phase levels to form a simple MDS code, whereas, in the second one, in-phase and quadrature components of codeword elements are used to construct the MDS code. We depict practical schemes for using the proposed concepts with orthogonal frequency division multiplexing (OFDM). We analyze the performance in terms of the minimum Euclidean distance and bit error rate. We also show that the proposed techniques exhibit desirable properties such as efficient low-complexity detection, very simple bits-to-symbols, and symbols-to-bits mappings, and a better error performance when compared to the OFDM-IM and related schemes. More importantly, contrary to the vast majority of IM studies that focus on showing the superiority of the IM techniques against conventional modulation techniques, we show that modulation concepts based on a well-known MDS code can achieve better error performance than the IM and related schemes while exhibiting a structure as simple as these schemes.

Linear codes are considered. A code is characterized by the length n, the dimension k, and the minimum distance d. An (n,k,d) code over the finite field GF(q) is said to be maximum distance separable (MDS) if d=n-k+1. A. Krishna and D.V. Sarwate (1990) investigated the existence of pseudo-cyclic MDS codes over GF(q) of length n, where n divides q-1 or q+1. It is shown that the pseudo-cyclic MDS codes constructed by Krishna and Sarwate are generalised Reed-Solomon codes. Pseudo-cyclic codes are studied over GF(q), where n and q=p/sup h/ are not relatively prime. It is proven that pseudo-cyclic (n,k) MDS codes modulo (x/sup n/-a) over GF(q) exist, if and only if n= p. Furthermore, any pseudo-cyclic (p,k) code modulo (x/sup p/-a) over GF(q) turns out to be MDS and generalized Reed-Solomon. It is explicitly proven that some classes of pseudo-cyclic (n,k) MDS codes over GF(q) are generalized Reed-Solomon codes. Furthermore, pseudo-cyclic (q+1,4) MDS codes over GF(q), q even, are completely classified. >

In this paper, we consider a novel wireless network for content distribute storage, where devices could use device-to-device (D2D) communication to transmit data. Several storage schemes, such as maximum distance separable (MDS) code and regenerating code, are employed for content downloading and repairing as node left and data lost. On the basis of multiparameter analysis, we found that targeting on the popular file caching, smaller download locality is the most suitable to the system based on fixed storage nodes and lower repair frequency. Moreover, MBR codes do not apply to D2D distributed storage due to high download cost, and MDS code is a convenient and appropriate scheme for majority practical D2D networks.

Determining the insdel distance of linear codes is a very challenging problem. Recently, Ji et al. [‘Strict half-Singleton bound, strict direct upper bound for linear insertion-deletion codes and optimal codes’, IEEE Trans. Inform. Theory 69 (5) (2023), 2900–2910] proposed their strict half-Singleton bound for linear codes that do not include the all-one vector. A natural question asks for linear codes with both large Hamming distance and insdel distance. Almost maximum distance separable (MDS) codes are a special kind of linear code with good Hamming error-correcting capability. We present a sufficient condition for the insdel distance of almost MDS codes to be bounded by the strict half-Singleton bound. In addition, we construct a class of two-dimensional near MDS codes that achieve the strict half-Singleton bound using twisted Reed–Solomon codes. Finally, we present a construction of optimal almost MDS insdel codes from Reed–Solomon codes for large dimensions.

The algebraic setting for threshold secret sharing scheme can vary, dependent on the application. This algebraic setting can limit the number of participants of an ideal secret sharing scheme. Thus it is important to know for which thresholds one could utilize an ideal threshold sharing schemeand for which thresholds one would have to use non-ideal schemes. The implication is that more than one share may have to be dealt to some or all parties. Karnin, Greene and Hellman constructed several bounds concerning the maximal number of participants in threshold sharing scheme. There has been a number of researchers who have noted the relationship between k-arcs in projective spaces and ideal linear threshold secret schemes, as well as between MDS codes and ideal linear threshold secret sharing schemes. Further, researchers have constructed optimal bounds concerning the size of k-arcs in projective spaces, MDS codes, etc. for various finite fields. Unfortunately, the application of these results on the Karnin, Greene and Hellamn bounds has not been widely disseminated. Our contribution in this paper is revisiting and updating the Karnin, Greene, and Hellman bounds, providing optimal bounds on the number of participants in ideal linear threshold secret sharing schemes for various finite fields, and constructing these bounds using the same tools that Karnin, Greene, and Hellman introduced in their seminal paper. We provide optimal bounds for the maximal number of players for a tout of nideal linear threshold scheme when t= 3, for all possible finite fields. We also provide bounds for infinitely many tand infinitely many fields and a unifying relationship between this problem and the MDS (maximum distance separable) codes that shows that any improvement on bounds for ideal linear threshold secret sharing scheme will impact bounds on MDS codes, for which there is a number of conjectured (but open) problems.

Mobile edge caching, that prefetches the frequently requested contents at edge nodes, is envisaged to play the leading role in reducing network traffic and latency. Limited storage space at each edge node necessitates efficient caching strategies of what and how to store in advance, and storing segmented content based on maximum distance separable (MDS) code enables to utilize the limited storage space more efficiently. This paper designs a strategy of probabilistic content caching based on MDS code to minimize the help from a macro base station and thus to lessen backhaul traffic. Desirable length of an MDS coded segment and the probability of caching an MDS coded segment are determined to maximize the amount of information obtained from neighboring edge nodes. Numerical results demonstrate that the proposed caching strategy enable to lessen backhaul load effectively.

The increasing expansion of data scale leads to the widespread deployment of storage systems with larger capacity and further induces the climbing probability of data loss or damage. The Maximum Distance Separable (MDS) code in RAID-6, which tolerates the concurrent failures of any two disks with minimal storage requirement, is one of the best candidates to enhance the data reliability. However, most of the existing works in this literature are more inclined to be specialized and cannot provide a satisfied performance under an all-round evaluation. Aiming at this problem, we propose an all-round MDS code named Horizontal-Vertical Code (HV Code) by taking advantage of horizontal parity and vertical parity. HV Code achieves the perfect I/O balancing and optimizes the operation of partial stripe writes to continuous data elements, while preserving the optimal encode/decode/update efficiency. Moreover, it owns a shorter parity chain which grants it a more efficient recovery for one disk failure. HV Code also behaves well for the degraded read operation and accelerates the process to reconstruct two disabled disks by executing four recovery chains in parallel. The performance evaluation demonstrates that HV Code well balances the I/O distribution and eliminates up to 27.6% and 32.4% I/O requests for partial stripe writes operation when compared with RDP Code and HDP Code. Moreover, compared to RDP Code, HDP Code, X-Code and H-Code, HV Code reduces up to 5.4%~39.8% I/O requests per element for the single disk reconstruction, decreases 6.6%~28.3% I/O requests for degraded read operations, and achieves the same efficiency of X-Code for double disk recovery by shortening 47.4%~59.7% recovery time compared with other three codes.

In this paper, we first construct several classes of classical Hermitian self-orthogonal maximum distance separable (MDS) codes. Through these classical codes, we are able to obtain various quantum MDS codes. It turns out that many of our quantum codes are new in the sense that the parameters of our quantum codes cannot be obtained from all previous constructions.

A $q$-ary code of length $n$, size $M$, and minimum distance $d$ is called an $(n,M,d)_q$ code. An $(n,q^{k},n-k+1)_q$ code is called a maximum distance separable (MDS) code. In this work, some MDS codes over small alphabets are classified. It is shown that every $(k+d-1,q^k,d)_q$ code with $k\geq 3$, $d \geq 3$, $q \in \{5,7\}$ is equivalent to a linear code with the same parameters. This implies that the $(6,5^4,3)_5$ code and the $(n,7^{n-2},3)_7$ MDS codes for $n\in\{6,7,8\}$ are unique. The classification of one-error-correcting $8$-ary MDS codes is also finished; there are $14$, $8$, $4$, and $4$ equivalence classes of $(n,8^{n-2},3)_8$ codes for $n=6,7,8,9$, respectively. One of the equivalence classes of perfect $(9,8^7,3)_8$ codes corresponds to the Hamming code and the other three are nonlinear codes for which there exists no previously known construction.

For ultra-dense networks with wireless backhaul, caching strategy at small base stations (SBSs), usually with limited storage, is critical to meet massive high data rate requests. Since the content popularity profile varies with time in an unknown way, we exploit reinforcement learning (RL) to design a cooperative caching strategy with maximum-distance separable (MDS) coding. We model the MDS coding based cooperative caching as a Markov decision process to capture the popularity dynamics and maximize the long-term expected cumulative traffic load served directly by the SBSs without accessing the macro base station. For the formulated problem, we first find the optimal solution for a small-scale system by embedding the cooperative MDS coding into Q-learning. To cope with the large-scale case, we approximate the state-action value function heuristically. The approximated function includes only a small number of learnable parameters and enables us to propose a fast and efficient action-selection approach, which dramatically reduces the complexity. Numerical results verify the optimality/near-optimality of the proposed RL based algorithms and show the superiority compared with the baseline schemes. They also exhibit good robustness to different environments.

The diffusion layer of the SPN block ciphers is usually built on the basis of the MDS (Maximum Distance Separable) matrices which is the matrix of the maximum distance separable code (MDS code). MDS codes have long been studied in error correcting code theory and have applications not only in coding theory but also in the design of block ciphers and hash functions. Thanks to that important role, there have been many studies on methods of building MDS matrices. In particular, the recursive MDS matrices and the symmetric recursive MDS matrices have particularly important applications because they are very efficient for execution. In this paper, we will give an estimate of the number of MDS matrices, recursive MDS matrices and symmetric recursive MDS matrices built from Reed-Solomon codes. This result is meaningful in determining the efficiency from this method of building matrices based on the Reed-Solomon codes. From there, this method can be applied to find out many MDS matrices, secure and efficient symmetric recursive MDS matrices for execution to apply in current block ciphers. Furthermore, recursive MDS matrices can be efficiently implemented using Linear Feedback Shift Registers (LFSR), making them well suited for lightweight cryptographic algorithms, so suitable for limited resources application.

Maximum distance separable (MDS) codes and near MDS (NMDS) codes play a fundamental role in coding theory due to their maximal error-correcting capabilities and rich algebraic structure. This paper constructs a new class of linear codes by twice extending TGRS codes, denoted as $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $. First, we establish the necessary and sufficient conditions for $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ to be MDS. By analyzing the AMDS properties of both $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ and their duals $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty)^{\bot} $, we determine the necessary and sufficient conditions for $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ to be NMDS. Besides, we provide the form of the parity check matrix for $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ and characterize the weight distributions of both the codes $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ and their duals. Additionally, using the Schur method, the non-generalized Reed-Solomon (non-GRS) properties are proven. Finally, several classes of linear complementary dual (LCD) codes are constructed.

An embedding of a code is a mapping that preserves distances between codewords. We prove that any code with code distance and length can be embedded into an maximum distance separable (MDS) code with the same code distance and length but under a larger alphabet. As a corollary we obtain embeddings of systems of partial mutually orthogonal Latin cubes and ‐ary quasigroups.