What makes a good code?

The design criteria for space-time codes under block fading channels are the rank criterion and the determinant criterion, which are widely cited in the literature. These criteria are not very tight under some circumstances. In this letter, we derive a tighter design criterion for space-time codes. Based on our criterion, we have found good space-time codes by computer search. The complexity of systematic search for good space-time codes is a big obstacle to get good codes. Using our new design criterion, the complexity of search is dropped down dramatically. Simulations show that our new codes have superior performance over the existing codes.

This paper presents a new method of constructing good binary linear codes. The proposed method is first based on the BCH codes parity matrix and Hadamard’s matrix, and after on a GUAVA-based computer search, which allowed us to find several good codes with rates >0.5.

Two compressing construction methods of Quasi-Cyclic LDPC (QC-LDPC) codes are proposed, which can be called recursive-compressing and single-compressing respectively. Selecting a good QC-LDPC code as the Mother, the son codes constructed through the compressing methods are still good codes, i.e., have high girth and few short cycles, moreover, they have continuous code length and require smaller memory space. Analysis and simulation results show they can perform well over the AWGN channel.

The weight spectrum of sequences of binary linear codes that achieve arbitrarily small word error probability on a class of noisy channels at a nonzero rate is studied. We refer to such sequences as good codes. The class of good codes includes turbo, low-density parity-check, and repeat-accumulate codes. We show that a sequence of codes is good when transmitted over a memoryless binary-symmetric channel (BSC) or an additive white Gaussian noise (AWGN) channel if and only if the slope of its spectrum is finite everywhere and its minimum Hamming distance goes to infinity with no requirement on its rate growth. The extension of these results to code ensembles in probabilistic terms follows in a direct manner. We also show that the sufficient condition holds for any binary-input memoryless channel.

In this paper, we investigate the behavior of the distance spectrum of convolutional codes when the decoding complexity is measured by the total number of edge symbols per information bit in the minimal trellis module representing the code. We conduct a code search restricted to the recently introduced class of generalized punctured convolutional codes, which is broad enough to contain good codes and yet has structural properties that facilitate the code search. New good convolutional codes are tabulated. For the same decoding complexity and the same code rate, the new codes have slightly better distance spectrum than the best known punctured convolutional codes. When compared to the best known unit-memory convolutional codes of the same rate, the new codes typically have improved free distance for a given decoding complexity, or about the same distance spectrum is achieved with much lower decoding complexity. Under this decoding complexity measure, the behaviour that higher decoding complexity implies better distance spectrum remains valid.

A new approach to constructing good long trellis codes for use with Sequential Decoding (SD) is proposed. The procedure begins by choosing a relatively small set of codes randomly. The error performance of each of these codes is evaluated using sequential decoding and the code with the best performance among the chosen set is retained. The performance of many of the randomly chosen codes is quite good. This is consistent with the well known fact that a randomly chosen code is very likely to be a good code. It is surprising to find out that the new codes found using this approach, which come from a very small set of codes compared to the total number of possible codes, perform about as well as the best known codes. The best short codes can be found by exhaustive search. This is the approach used to construct codes for Viterbi decoding, whose complexity prohibits the use of long codes. On the other hand, the complexity of SD is essentially independent of the code constraint length. The approach proposed here thus provides an excellent way of constructing long codes for use with SD, since an exhaustive search for the best long codes is impractical.

We study the average error probability performance of binary linear code ensembles when each codeword is divided into J subcodewords with each being transmitted over one of J parallel channels. This model is widely accepted for a number of important practical channels and signaling schemes including block-fading channels, incremental redundancy retransmission schemes, and multicarrier communication techniques for frequency-selective channels. Our focus is on ensembles of good codes whose performance in a single channel model is characterized by a threshold behavior, e.g., turbo and low-density parity-check (LDPC) codes. For a given good code ensemble, we investigate reliable channel regions which ensure reliable communications over parallel channels under maximum-likelihood (ML) decoding. To construct reliable regions, we study a modifed 1961 Gallager bound for parallel channels. By allowing codeword bits to be randomly assigned to each component channel, the average parallel-channel Gallager bound is simplified to be a function of code weight enumerators and channel assignment rates. Special cases of this bound, average union-Bhattacharyya (UB), Shulman-Feder (SF), simplified-sphere (SS), and modified Shulman-Feder (MSF) parallel-channel bounds, allow for describing reliable channel regions using simple functions of channel and code spectrum parameters. Parameters describing the channel are the average parallel-channel Bhattacharyya noise parameter, the average channel mutual information, and parallel Gaussian channel signal-to-noise ratios (SNRs). Code parameters include the union-Bhattacharyya noise threshold and the weight spectrum distance to the random binary code ensemble. Reliable channel regions of repeat-accumulate (RA) codes for parallel binary erasure channels (BECs) and of turbo codes for parallel additive white Gaussian noise (AWGN) channels are numerically computed and compared with simulation results based on iterative decoding. In addition, an examp

The workshop Finite Fields: Theory and Applications was organized by Joachim von zur Gathen (Bonn), Igor Shparlinski (Sydney), and Henning Stichtenoth (Essen), and ran from 5 to 11 December 2004. Its forty participants, with a wide geographical distribution, enjoyed the hospitality of the Mathematical Research Institute, and its beautiful surroundings. Two previous meetings on the topic had been held in 1997 and 2001. The schedule consisted of three plenary talks each morning, and specialized sessions later in the day, with vast time for discussions and collaborative work. The traditional Wednesday afternoon hike was blessed with wonderful sunny weather and the compulsory Black Forest cake reward at the end. Very broadly, we can distinguish seven subject areas: Of course, many of the results presented bridge between two or more of these areas. The abstracts that follow speak for themselves. Avoiding an exhaustive discussion, we now mention one particular talk from each of the seven areas. The structure theory includes questions about polynomials. The well-known Hansen–Mullen conjecture (whose second author was in the audience) was stated in 1992 and asserts that for any finite field \mathbb F_{q} , integers n and m with 0 < m < n and a \in \mathbb F_{q} , there exists a monic primitive polynomial in \mathbb F_{q}[x] of degree n having a as the coefficient of x^{m} ; there are a few well-known exceptional cases where this fails to hold. Cohen presented a proof of this conjecture at degrees n \ge 9 , assuring the audience that smaller values of n are also under consideration. Towers of function fields are of great interest because they may yield good algebraic-geometric codes. Beelen introduced a recursive construction of such towers, using a certain type of Fuchsian differential equations. They can be obtained from modular curves, and in some cases can be shown to be asymptotically optimal (in terms of the parameters of the resulting codes). A conjecture concerning points on varieties was stated by Heath–Brown. Namely, he considers a nonsingular nonlinear hypersurface X in \mathbb P^{n} defined over \mathbb Q , considers the number N(B) of points on X with rational integral coefficients absolutely bounded by B , and conjectures that this number is O(B^{n-1+\epsilon}) for any positive \epsilon . Browning presented his proof of this conjecture in all cases, with the possible exceptions d = 3,4 and n = 7,8 . In the theory of error-correcting codes , finite fields were fundamental from its beginning in the 1940s. Their importance was heightened by the construction of codes from algebraic curves over finite fields. Voloch discussed a different connection: the quadratic residue codes. It is unknown whether subfamilies of them can yield asymptotically good codes. Voloch showed that there exist subfamilies that do not yield good codes. This is based on an expression of the minimal distance by exponential sums, due to Helleseth, and estimates on the smallest prime that splits completely in a number field. For computation , a difficult class of objects are bivariate polynomials presented in a particularly generous format, namely as a sum of terms where the exponents are written in binary (or decimal). Thus we look at polynomials of humongous degrees. Kaltofen presented two results which illuminate the wide range of behavior for questions about such polynomials. Over the rational numbers, he can compute the linear and quadratic factors in polynomial time. Over a large finite field, testing irreducibility is NP-hard (under randomized reductions). As a question from combinatorics , we give the following illustrative example. A sum-free set A in an additive group G is such that x+y \not= z for all x, y, z \in A . For instance the additive group G = \mathbb Z_{p} for a prime p and A=\{n, n+1, \ldots, 2n-1\} for n= \lfloor (p+1)/3 \rfloor is a sum-free set. We can also multiply each element of A by a fixed nonzero element of \mathbb Z_{p} . When p \equiv 2 \bmod 3 , no other sum-free subsets of \mathbb Z_{p} exist. Lev shows that assumption \#A \geq 0.33p implies that A is contained in the corresponding interval or a dilation of it. In cryptography , a central question is the conjectured difficulty of computing the discrete logarithm in certain groups. The method of index calculus provides a subexponential algorithm in the unit groups of finite fields. Elliptic curves owe their popularity in cryptography to the absence, so far, of any discrete logarithm computation of comparable efficiency. Semaev presented an approach, rather speculative at this point, aimed at finding such a method; it works with the new notion of summation polynomials which vanish at the x -coordinates of points that sum to 0 on the curve.

In this paper, we shall give an explicit proof that constacyclic codes over finite commutative rings can be realized as ideals in some twisted group rings. Also, we shall study isometries between those codes and, finally, we shall study k-Galois LCD constacyclic codes over finite fields. In particular, we shall characterize constacyclic LCD codes with respect to Euclidean inner product in terms of its idempotent generators and the classical involution using the twisted group algebras structures and find some good LCD codes

One of the most important and challenging problems in coding theory is to construct codes with the best possible parameters. Quasi-cyclic (QC) and the larger class of quasi- twisted (QT) codes have been proven to contain many good codes (with best-known parameters). In this paper, we review some open problems concerning these codes, introduce generalizations of QT codes, and suggest some constructions involving QT codes. We also present some new and good quaternary codes.

Multiple input/multiple output (MIMO) radars [1] and orthogonal netted radar systems (ONRS) [2], [3] use spatial diversity to fundamentally improve radar performance. To realize their full potential in performance, they require a “good” code set. Each radar station in a MIMO radar or an ONRS transmits its own good code, belonging to the set of good codes. On reception, the radar stations process their signal returns as well as signal returns from other radar stations and the resulting data is then fed into a fusion processor. The fusion processor uses the data from each of the radar stations in the MIMO radar/ONRS to form an elaborate 3D image of the area of interest.

We show how good quantum error-correcting codes can be constructed using generalized concatenation. The inner codes are quantum codes, the outer codes can be linear or nonlinear classical codes. Many new good codes are found, including both stabilizer codes as well as so-called nonadditive codes.

Multi-twisted (MT) codes were introduced as a generalization of quasi-twisted (QT) codes. QT codes have been known to contain many good codes. In this work, we show that codes with good parameters and desirable properties can be obtained from MT codes. These include best known and optimal classical codes with additional properties such as reversibility and self-duality, and new and best known non-binary quantum codes obtained from special cases of MT codes. Often times best known quantum codes in the literature are obtained indirectly by considering extension rings. Our constructions have the advantage that we obtain these codes by more direct and simpler methods. Additionally, we found theoretical results about binomials over finite fields that are useful in our search.

In this work we give two new constructions of e-biased generators. Our first construction significantly extends a result of Mossel et al. (Random Structures and Algorithms 2006, pages 56-81), and our second construction answers an open question of Dodis and Smith (STOC 2005, pages 654-663). In particular we obtain the following results: For every k = o(log n) we construct an e-biased generator $$G : \{0, 1\}^{m} \rightarrow \{0, 1\}^n$$that is implementable by degree k polynomials (namely, every output bit of the generator is a degree k polynomial in the input bits). For any constant k we get that $$n = \Omega(m/{\rm log}(1/ \epsilon))^k$$, which is nearly optimal. Our result also separates degree k generators from generators in NC0k, showing that the stretch of the former can be much larger than the stretch of the latter. The problem of constructing degree k generators was introduced by Mossel et al. who gave a construction only for the case of k = 2. We construct a family of asymptotically good binary codes such that the codes in our family are also e-biased sets for an exponentially small e. Our encoding algorithm runs in polynomial time in the block length of the code. Moreover, these codes have a polynomial time decoding algorithm. This answers an open question of Dodis and Smith. The paper also contains an appendix by Venkatesan Guruswami that provides an explicit construction of a family of error correcting codes of rate 1/2 that has efficient encoding and decoding algorithms and whose dual codes are also good codes.

Justesen presented the asymptotically good block codes, and then the asymptotically good convolutional codes R j in a constructive way. The latter are the concatenated codes, where the outer code is the convolutional code generated by the generator polynomial of Reed‐Solomon code, and the inner code is Wozencraft's randomly shifted code. The performance of Justesen's convolutional code is not necessarily high for low code rates. Sugiyama, Kasahara, Hirasawa and Namekawa presented new asymptotically good convolutional codes R c by applying the second order concatenation to Justesen's convolutional codes R j , which gave an improvement for the low code rates. This paper presents constructively new asymptotically good convolutional codes R ( J ) , by comgining the second‐order concatenation of the forementioned improved convolutional codes R c and the generalized concatenated codes. It is shown that the new codes R ( J ) always have a higher performance than that of the convolutional codes R c , and have the best performance for the low code rates 0< R <0.25., among the asymptotically good convolutional codes known up to now.