Large Minimum Distance Research Articles

A Random Local Matroid Search Algorithm to Construct Good Rate <inline-formula> <tex-math notation="TeX">$1/p$</tex-math></inline-formula> Systematic Binary Quasi-Cyclic Codes

In this letter, a random local matroid search algorithm is proposed to construct good rate 1/p systematic QC codes based on the relationship between the generator matrix and the minimum distance when the length of the information vector is k ≥ 15 . More than twenty new QC codes are obtained and their corresponding generator matrices are revealed, which extends previous results. Nine of these new codes have larger minimum distances than the reported ones in the previous literature.

Application of Constacyclic Codes to Quantum MDS Codes

Quantum maximal-distance-separable (MDS) codes form an important class of quantum codes. To get $q$-ary quantum MDS codes, it suffices to find linear MDS codes $C$ over $\mathbb{F}_{q^2}$ satisfying $C^{\perp_H}\subseteq C$ by the Hermitian construction and the quantum Singleton bound. If $C^{\perp_{H}}\subseteq C$, we say that $C$ is a dual-containing code. Many new quantum MDS codes with relatively large minimum distance have been produced by constructing dual-containing constacyclic MDS codes (see \cite{Guardia11}, \cite{Kai13}, \cite{Kai14}). These works motivate us to make a careful study on the existence condition for nontrivial dual-containing constacyclic codes. This would help us to avoid unnecessary attempts and provide effective ideas in order to construct dual-containing codes. Several classes of dual-containing MDS constacyclic codes are constructed and their parameters are computed. Consequently, new quantum MDS codes are derived from these parameters. The quantum MDS codes exhibited here have parameters better than the ones available in the literature.

New quantum MDS codes derived from constacyclic codes

Quantum maximum-distance-separable (MDS) codes form an important class of quantum codes. It is very hard to construct quantum MDS codes with relatively large minimum distance. In this paper, based on classical constacyclic codes, we construct two classes of quantum MDS codes with parameters $$\begin{aligned}{}[[\lambda (q-1),\lambda (q-1)-2d+2,d]]_q \end{aligned}$$ where $$2\le d\le (q+1)/2+\lambda -1$$ , and $$q+1=\lambda r$$ with $$r$$ even, and $$\begin{aligned}{}[[\lambda (q-1),\lambda (q-1)-2d+2,d]]_q \end{aligned}$$ where $$2\le d\le (q+1)/2+\lambda /2-1$$ , and $$q+1=\lambda r$$ with $$r$$ odd. The quantum MDS codes exhibited here have parameters better than the ones available in the literature.

Entanglement-assisted quantum codes achieving the quantum singleton bound but violating the quantum hamming bound

We give an infinite family of degenerate entanglement-assisted quantum error-correcting codes (EAQECCs) which violate the EA-quantum Hamming bound for non-degenerate EAQECCs and achieve the EA-quantum Singleton bound, thereby proving that the EA-quantum Hamming bound does not asymptotically hold for degenerate EAQECCs. Unlike the previously known quantum error-correcting codes that violate the quantum Hamming bound by exploiting maximally entangled pairs of qubits, our codes do not require local unitary operations on the entangled auxiliary qubits during encoding. The degenerate EAQECCs we present are constructed from classical error-correcting codes with poor minimum distances, which implies that, unlike the majority of known EAQECCs with large minimum distances, our EAQECCs take more advantage of degeneracy and rely less on the error correction capabilities of classical codes.

On the Optimum Cyclic Subcode Chains of <inline-formula> <tex-math notation="TeX">${\cal RM}(2, m)^{\ast}$</tex-math></inline-formula> for Increasing Message Length

The distance profiles of linear block codes can be employed to design variational coding scheme for encoding message with variational length and getting lower decoding error probability by large minimum Hamming distance, where one example is in the design of transport format combination indicators (TFCIs) in CDMA. Considering convenience for encoding, we focus on the distance profiles with respect to cyclic subcode chains (DPCs) of cyclic codes over GF(q) with length n such that gcd(n, q) = 1. In this paper, the optimum DPCs and the corresponding optimum cyclic subcode chains are investigated on the punctured second-order Reed-Muller code RM(2, m)* for increasing message length, where two standards on the optimums are studied according to the rhythm of increase. Ignoring the dimension profile, the device will coincide with that of TFCI.

A Construction of New Quantum MDS Codes

It has been a great challenge to construct new quantum maximum-distance-separable (MDS) codes. In particular, it is very hard to construct the quantum MDS codes with relatively large minimum distance. So far, except for some sparse lengths, all known q-ary quantum MDS codes have minimum distance ≤q/2 + 1. In this paper, we provide a construction of the quantum MDS codes with minimum distance >q/2 + 1. In particular, we show the existence of the q-ary quantum MDS codes with length n = q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> + 1 and minimum distance d for any d q + 1 (this result extends those given in the works of Guardia (2011), Jin et al. (2010), and Kai an Zhu (2012)); and with length (q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> + 2)/3 and minimum distance d for any d (2q+2)/3 if 3|(q + 1). Our method is through Hermitian selforthogonal codes. The main idea of constructing the Hermitian self-orthogonal codes is based on the solvability in F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> of a system of homogenous equations over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> .

Constacyclic Codes and Some New Quantum MDS Codes

One central theme in quantum error-correction is to construct quantum codes that have a large minimum distance. Quantum maximal distance separable (MDS) codes are optimal in the sense they attain maximal minimum distance. Recently, constructing quantum MDS codes has received much attention and seems to become more and more difficult. In this paper, based on classical constacyclic codes, we construct some new quantum MDS codes by employing the Hermitian construction. Compared with the known quantum MDS codes, these quantum MDS codes have much larger minimum distance.

Flexible Codebook Design for Limited Feedback Systems Via Sequential Smooth Optimization on the Grassmannian Manifold

Grassmannian quantization codebooks play a central role in a number of limited feedback schemes for single and multi-user multiple-input multiple-output (MIMO) communication systems. In practice, it is often desirable that these codebooks possess additional properties that facilitate their implementation, beyond the provision of good quantization performance. Although some good codebooks exist, their design tends to be a rather intricate task. The goal of this paper is to suggest a flexible approach to the design of Grassmannian codebooks based on sequential smooth optimization on the Grassmannian manifold and the use of smooth penalty functions to obtain additional desirable properties. As one example, the proposed approach is used to design rank-2 codebooks that have a nested structure and elements from a phase-shift keying (PSK) alphabet. In some numerical comparisons, codebooks designed using the proposed methods have larger minimum distances than some existing codebooks, and provide tangible performance gains when applied to a simple MIMO downlink scenario with zero-forcing beamforming, per-user unitary beamforming and rate control (PU 2RC), and block diagonalization signalling. Furthermore, the proposed approach yields codebooks that attain desirable additional properties without incurring a substantial degradation in performance.

Short Barcodes for Next Generation Sequencing

We consider the design and evaluation of short barcodes, with a length between six and eight nucleotides, used for parallel sequencing on platforms where substitution errors dominate. Such codes should have not only good error correction properties but also the code words should fulfil certain biological constraints (experimental parameters). We compare published barcodes with codes obtained by two new constructions methods, one based on the currently best known linear codes and a simple randomized construction method. The evaluation done is with respect to the error correction capabilities, barcode size and their experimental parameters and fundamental bounds on the code size and their distance properties. We provide a list of codes for lengths between six and eight nucleotides, where for length eight, two substitution errors can be corrected. In fact, no code with larger minimum distance can exist.

Sporadic neighbour-transitive codes in Johnson graphs

We classify the neighbour-transitive codes in Johnson graphs $$J(v,k)$$ of minimum distance at least three which admit a neighbour-transitive group of automorphisms that is an almost simple two-transitive group of degree $$v$$ and does not occur in an infinite family of two-transitive groups. The result of this classification is a table of 22 codes with these properties. Many have relatively large minimum distance in comparison to their length $$v$$ and number of code words. We construct an additional five neighbour-transitive codes with minimum distance two admitting such a group. All 27 codes are $$t$$ -designs with $$t$$ at least two.

A note on binary completely regular codes with large minimum distance

We classify all binary error correcting completely regular codes of length n with minimum distance δ>n/2.

Long Irregular LDPC coded OFDM with Soft Decision

OFDM with quadrature amplitude modulation (QAM) technique can be used for high speed optical applications. Increasing the order of modulation, the bit error rate (BER) increases. Forward Error correction (FEC) coding like LDPC coding is generally used to improve the BER performance. LDPC provides large minimum distance and also the power efficiency of the LDPC code increases significantly with the code length. In this paper we have compared the Soft decision and Hard decision algorithms of LDPC codes. A long Irregular LDPC code is simulated over the BIAWGN channel demonstrating the fact that LDPC coded OFDM with soft decision decoding provides very lower bit error rate as well as a larger gain in transmitter power and thus making the link more power efficient than the case with hard decision decoding. Through simulation, we have shown the advantages of using this long Irregular LDPC coded OFDM link in Optical wireless communication (OWC).

Optimal and isodual ternary cyclic codes of rate 1/2

This work is twofold. First, the largest minimum distance of a ternary cyclic codes of parameters (n, n 2 ), is determined for n even, not a multiple of 3, by using the Chen algorithm, for n = 26, 34, 38, 46, 50, 58, 62, 68, 70, 74. Next, seven new classes of isodual ternary cyclic codes are introduced for n singly even, not a multiple of 3.

On the Minimum Distance of Turbo Codes With Quadratic Permutation Polynomial Interleavers

An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are of particular interest because they admit analytical designs and simple, practical hardware implementations. Also, the recently proposed quadratic permutation polynomial (QPP)-based interleavers by Sun and Takeshita have provided excellent performance for short-to-medium block lengths, and have been selected for the 3GPP LTE standard. In this paper, we derive some upper bounds on the best achievable minimum distance dmin of QPP-based conventional binary turbo codes (with tailbiting termination, or dual termination when the interleaver length N is sufficiently large) that are tight for larger block sizes. In particular, we show that the minimum distance is at most 2(2v+1 + 9), independent of the interleaver length, when there exists an inverse polynomial of degree two, where v is the degree of the primitive feedback and monic feedforward polynomials. However, allowing the QPP to have no inverse polynomials of degree two may give strictly larger minimum distances (and lower multiplicities). In particular, we provide several QPPs with no quadratic inverse for some of the 3GPP LTE interleaver lengths giving a dmin with the 3GPP LTE constituent encoders which is strictly larger than 50. For instance, we have found a QPP for N = 6016 which gives an estimated dmin of 57. Furthermore, we provide the exact minimum distances and the corresponding multiplicities for all 3GPP LTE turbo codes (with dual termination) which shows that the best minimum distance is 51. Finally, we compute the best achievable minimum distances with QPP interleavers for all 3GPP LTE interleaver lengths N ≤ 4096, and compare these minimum distances with the ones we get when using the 3GPP LTE polynomials.

Constructing rate 1/p systematic binary quasi-cyclic codes based on the matroid theory

In this paper, rate 1/p binary systematic quasi-cyclic (QC) codes are constructed based on Matroid Theory (MT). The relationship between the generator matrix and minimum distance d is derived through MT, which is benefit to find numbers of QC codes with large minimum distance by our Matroid search algorithm. More than seventy of QC codes that extend previously published results are presented. Among these codes, there are nine codes whose minimum distance is larger than those of the known codes found by Gulliver et al.

Cyclic and Quasi-Cyclic LDPC Codes on Constrained Parity-Check Matrices and Their Trapping Sets

This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly low-density parity-check (LDPC) codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descendant codes are developed, including the characterization of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite-geometry LDPC codes using the results developed in the first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping set structure of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite-geometry and finite-field cyclic and quasi-cyclic LDPC codes with large minimum distances are shown to have no harmful trapping sets of size smaller than their minimum distances. Consequently, their error-floor performances are dominated by their minimum distances.

Nonbinary Cyclic LDPC Codes Derived from Idempotents and Modular Golomb Rulers

Based jointly on idempotents and modular Golomb rulers, we construct a class of nonbinary cyclic low-density parity-check (LDPC) codes. The defining parity-check matrix is a sparse circulant, on which we put two constraints: 1) the characteristic polynomial is an idempotent, 2) the nonzero elements of the first row are located on a modular Golomb ruler. We show that the second constraint forms a necessary and sufficient condition for the Tanner graph to have no cycles of length 4. The minimum distance of the code is proved equal to the column weight of the parity-check matrix plus one. A search algorithm is presented, with which we obtain some high rate codes with large minimum distances. The issue of code equivalence is also discussed. Simulation results show that the obtained codes perform well under iterative decoding.

Quasi-Cyclic LDPC Codes on Cyclic Subgroups of Finite Fields

A new class of quasi-cyclic LDPC codes whose parity-check matrices are arrays of circulant permutation matrices constructed based on cyclic subgroups of finite fields is presented. This class of codes contains several known classes of algebraic quasi-cyclic LDPC codes as subclasses. Experimental results show that the codes constructed perform very well over the AWGN channel when decoded with iterative decoding based on belief propagation. This class of new QC-LDPC codes contains a subclass of codes which have large minimum distances. Combinatorial expressions for the ranks of the parity-check matrices of a subclass of codes constructed based on fields of characteristic two are given.

Iterative Algorithms for Decoding a Class of Two-Step Majority-Logic Decodable Cyclic Codes

Codes constructed based on finite geometries form a large class of cyclic codes with large minimum distances which can be decoded with simple majority-logic decoding in one or multiple steps. In 2001, Kou, Lin and Fossorier showed that the one-step majority-logic decodable finite geometry codes form a class of cyclic LDPC codes whose Tanner graphs are free of cycles of length 4. These cyclic finite geometry LDPC codes perform very well over the AWGN channel using iterative decoding based on belief propagation (IDBP) and have very low error-floors. However, the standard IDBP is not effective for decoding other cyclic finite geometry codes because their Tanner graphs contain too many short cycles of length 4 which severely degrade the decoding performance. This paper investigates iterative decoding of two-step majority-logic decodable finite geometry codes. Three effective algorithms for decoding these codes are proposed. These algorithms are devised based on the orthogonal structure of the parity-check matrices of the codes to avoid or reduce the degrading effect of the short cycles of length 4. These decoding algorithms provide significant coding gains over the standard IDBP using either the sum-product or the min-sum algorithms.

Binary Grassmannian Weightbooks for MIMO Beamforming Systems

Binary weightbooks for multiple-input multiple-output (MIMO) beamforming systems using quantized feedback are designed based on the Grassmannian beamforming criterion. It is shown that the Grassmannian criterion for binary weightbook design is to maximize the minimum Hamming distance of the corresponding block code. Thus, block codes that have large minimum Hamming distance can be used for binary weightbook design for MIMO beamforming systems using quantized feedback. The optimum binary weightbooks when the number of transmit antennas has value from two to six are tabulated.