Matrix-Product Codes over ? q

On dual-containing, almost dual-containing matrix-product codes and related quantum codes

Many classical constructions, such as Plotkin’s and Turyn’s, were generalized by matrix product (MP) codes. Quasi-twisted (QT) codes, on the other hand, form an algebraically rich structure class that contains many codes with best-known parameters. We significantly extend the definition of MP codes to establish a broader class of generalized matrix product (GMP) codes that contains QT codes as well. We propose a generator matrix formula for any linear GMP code and provide a condition for determining the code size. We prove that any QT code has a GMP structure. Then we show how to build a generator polynomial matrix for a QT code from its GMP structure, and vice versa. Even though the class of QT codes contains many codes with best-known parameters, we present different examples of GMP codes with best-known parameters that are neither MP nor QT. Two different lower bounds on the minimum distance of GMP codes are presented; they generalize their counterparts in the MP codes literature. The second proposed lower bound replaces the non-singular by columns matrix with a less restrictive condition. Some examples are provided for comparing the two proposed bounds, as well as showing that these bounds are tight.

Matrix product (MP) codes over a finite field [Formula: see text], where [Formula: see text] a prime power, were first introduced by Blackmore and Norton [MP codes over [Formula: see text], Appl. Algebra Engrg. Comm. Comput. 12 (2001) 477–500]. This paper investigates linear complementary dual (LCD) MP codes over finite commutative Frobenius rings. We derive a necessary and sufficient condition for an MP code to have an LCD. We provide some efficient constructions of LCD MP codes. Finally, we have shown an application on cryptography of LCD codes.

We consider certain subcodes of generalized Reed-Muller (GRM) codes, which we call homogeneous generalized Reed-Muller (HRM) codes. In general, they have a much better minimum distance than the GRM codes. The parameters of HRM codes are related to those of projective Reed-Muller (PRM) codes. Unlike most PRM codes, punctured HRM codes are cyclic. Under the trace map, HRM codes map to binary codes. These are in general much larger than classical RM codes, for the same minimum distance.

In this work, we present linear exact repair schemes for one or two erasures in decreasing monomial-Cartesian codes (DM-CC), a family of codes which provides a framework for polar codes. In the case of two erasures, the positions of the erasures should satisfy a certain restriction. We present families of augmented Reed-Muller (ARM) and augmented Cartesian codes (ACar) which are families of evaluation codes obtained by strategically adding vectors to Reed-Muller and Cartesian codes, respectively. We develop repair schemes for one or two erasures for these families of augmented codes. Unlike the repair scheme for two erasures of DM-CC, the repair scheme for two erasures for the augmented codes has no restrictions on the positions of the erasures. When the dimension and base field are fixed, we give examples where ARM and ACar codes provide a lower bandwidth (resp., bitwidth) in comparison with Reed-Solomon (resp., Hermitian) codes. When the length and base field are fixed, we give examples where ACar codes provide a lower bandwidth in comparison with ARM. Finally, we analyze the asymptotic behavior when the augmented codes achieve the maximum rate.

In emerging storage technologies, the outputs of the channels consist of overlapping pairs of symbols. The errors are no longer individual symbols. Controlling them calls for a different approach. Symbol-pair codes have been proposed as a solution. The error-correcting capability of such a code depends on its minimum pair distance instead of the usual minimum Hamming distance. Longer codes can be conveniently constructed from known shorter ones by a matrix-product approach. The parameters of a matrix-product code can be determined from the parameters of the ingredient codes. We construct a new family of maximum distance separable (MDS) symbol-pair matrix-product codes. Codes which are permutation equivalent to matrix-product codes may have improved minimum pair distances. We present four new families of MDS symbol-pair codes and a new family of almost MDS symbol-pair codes. The codes in these five new families are permutation equivalent to matrix-product codes. Each of our five constructions identifies permutations that can increase the minimum pair distances. We situate the new families among previously known families of MDS symbol-pair codes to highlight the versatility of our matrix-product construction route.

In construction of bandwidth efficient multilevel block codes, Reed-Muller codes with two-level squaring construction have been found to be good candidates for component codes. On fading channels where codes with large minimum Hamming distances are required, Reed-Muller codes are very much favored due to high code rate and low decoding complexity, as well as large minimum Hamming distances. The authors present a class of new MPSK modulation codes by using Reed-Muller codes as component codes for Rayleigh fading channels. Two modified types of Reed-Muller codes are proposed to achieve trade-offs between code rate and decoder complexity. By using the minimum Hamming distance as the design criterion, new MPSK BCM codes are constructed. These codes outperform optimum TCM codes, with comparable decoder complexity and bandwidth efficiency. >

The well known Plotkin construction is, in the current paper, generalized and used to yield new families of ℤ2ℤ4-additive codes, whose length, dimension as well as minimum distance are studied. These new constructions enable us to obtain families of ℤ2ℤ4-additive codes such that, under the Gray map, the corresponding binary codes have the same parameters and properties as the usual binary linear Reed-Muller codes. Moreover, the first family is the usual binary linear Reed-Muller family.Keywordsℤ2ℤ4-Additive codesPlotkin constructionReed-Muller codesℤ2ℤ4-linear codes

A locally repairable code (LRC) is a [n, k, d] linear code with length n, dimension k, minimum distance d and locality r, which means that every code symbol can be repaired by at most r other symbols. LRCs have become an important candidate in distributed storage systems due to their relatively low I/O cost. An LRC is said to be optimal if its minimum distance meets one of the Singleton-like bounds. This paper considers the optimal constructions of LRCs with locality r = 1 and r = k - 1, which involves three types: r-local LRCs, (r, δ)-LRCs and LRCs with t-availability. Specifically, we first prove that the existence of an optimal LRC with locality r = 1 is equivalent to that of an MDS code with certain parameters. Thus we can completely characterize the three types of optimal LRCs with r = 1 based on some known constructions of MDS codes. Near MDS codes is a special class of sub-optimal linear code whose minimum distance d = n - k and the i-th generalized Hamming weight achieves the generalized Singleton bound for 2 ≤ i ≤ k. For r = k - 1, we have established the connections between optimal r-local LRCs/ LRCs with t-availability and near MDS codes. Such connections can help to construct optimal LRCs with r = k - 1 from some known classes of near MDS codes.

Due to the ideal autocorrelation property, Golay complementary sets (GCSs) can be applied to orthogonal frequency division multiplexing (OFDM) systems for peak-to-average power ratio (PAPR) reduction. With the additional ideal cross-correlation property, complete complementary codes (CCCs), which consist of mutually orthogonal GCSs, can be employed in code-division multiple access (CDMA) systems to eliminate the multiple-access interference. It has been shown that both GCSs and CCCs can be obtained from cosets of the first-order generalized Reed-Muller codes. In the thesis, a unified work to construct families of complementary sets and CCCs from cosets of the first-order generalized Reed-Muller codes is proposed. Besides generalizing some previous results on GCSs and CCCs, extensions of GCSs and CCCs which have some desirable (even though nonideal) autocorrelation and/or cross-correlation properties are investigated, namely, multiple-shift complementary sets (MSCSs), quasi-complementary sets (QCSs), and quasi complete complementary codes (QCCCs). The relationship between the families of GCSs and generalized Reed-Muller codes is first investigated in the thesis. Direct generic constructions of GCSs, MSCSs, and QCSs from cosets of the first-order generalized Reed-Muller codes are proposed. Upper bounds on PAPRs of families of GCSs are also exploited. Then constructions of CCCs and QCCCs from generalized Reed-Muller codes are provided. A novel application of the constructed QCCCs is proposed in the thesis to employ them as the preamble sequences for cell search in cell-based OFDM systems, due to their good auto-correlation and cross-correlation properties as well as low PAPR values. Simulation results show that the proposed QCCC-based preambles outperform the preambles employed in the WiMAX system, both in terms of PAPR and cell search performance.

Squares of matrix-product codes

In this paper, we provide a general form for sparse generator matrices of several families of Quasi-Cyclic Low-Density Parity-Check codes. Codes of this kind have a prominent role in literature and applications due to their ability to achieve excellent performance with limited complexity. While some properties of these codes (like the girth length in their associated Tanner graphs) are well investigated, estimating their minimum distance is still an open problem. By obtaining sparse generator matrices for several families of these codes, we prove that they are also Quasi-Cyclic Low-Density Generator Matrix codes, which is an important feature to reduce the encoding complexity, and provides a useful tool for the investigation of their minimum distance.

For a given finitely many codes, Matrix Product Code (MPC) can generate a code with better dimension and minimum distance. In this paper, SageMath functions are defined and using these functions, matrix product codes (MPC) are constructed with defining matrix as the generator matrix of some BHC-code or RS-code.

For a given finitely many codes, Matrix Product Code (MPC) can generate a code with better dimension and minimum distance. In this paper, SageMath functions are defined and using these functions, matrix product codes (MPC) are constructed with defining matrix as the generator matrix of some BHC-code or RS-code.

The Theory of Error-Correcting Codes