Codes From Algebraic Geometry

This paper studies linear and affine error-correcting codes for correcting synchronization errors such as insertions and deletions. We call such codes linear/affine insdel codes. Linear codes that can correct even a single deletion are limited to having an information rate at most (achieved by the trivial two fold repetition code). Previously, it was (erroneously) reported that more generally no nontrivial linear codes correcting deletions exist, i.e., that the -fold repetition codes and its rate of are basically optimal for any . We disprove this and show the existence of binary linear codes of length and rate just below capable of correcting insertions and deletions. This identifies rate as a sharp threshold for recovery from deletions for linear codes and reopens the quest for a better understanding of the capabilities of linear codes for correcting insertions/deletions. We prove novel outer bounds and existential inner bounds for the rate vs. (edit) distance trade-off of linear insdel codes. We complement our existential results with an efficient synchronization-string-based transformation that converts any asymptotically good linear code for Hamming errors into an asymptotically good linear code for insdel errors. Last, we show that the -rate limitation does not hold for affine codes by giving an explicit affine code of rate which can efficiently correct a constant fraction of insdel errors.

The most important development in the theory of error-correcting codes in recent years is the introduction of methods from algebraic geometry to construct good codes. The ideas are based on generalizations of so-called Goppa codes. The (by now) “classical” Goppa codes (1970, cf.[6]) were already a great improvement on codes known at that time. The algebraic geometric codes were also inspired by ideas of Goppa but the most sensational development was a paper by Tsfasman, Vlădut and Zink (1982,cf.[15]). In this paper the idea of codes from algebraic curves was combined with certain recent deep results from algebraic geometry to produce a sequence of error-correcting codes that led to a new lower bound on the information rate of good codes that is better than the Gilbert-Varshamov bound. The novice reader should realize that the G-V-bound (1952) was never improved (until 1982) and believed by many to be best possible. Actually, the improvement is only achieved for alphabets of size at least 49 and several binary coding experts still have hope that no improvement of the G-V-bound for F2 will be possible; (this author is not one of them).

Open problems in finite projective spaces

It has been well-known that the class of quasi-cyclic (QC) codes contain many good codes. In thispaper, a method to conduct a computer search for binary 2-generator QC codes is presented, anda large number of good 2-generator QC codes have been obtained. 5 new binary QC codes thatimprove the lower bounds on minimum distance are presented. Furthermore, with new 2-generatorQC codes and Construction X, 2 new improved binary linear codes are obtained. With the standardconstruction techniques, another 16 new binary linear codes that improve the lower bound on theminimum distance have also been obtained.

We introduce a new construction of error-correcting codes from algebraic curves over finite fields. Modular curves of genus g\ra\infty over a field of size q_0^2 yield nonlinear codes more efficient than the linear Goppa codes obtained from the same curves. These new codes now have the highest asymptotic transmission rates known for certain ranges of alphabet size and error rate. Both the theory and possible practical use of these new record codes require the development of new tools. On the theoretical side, establishing the transmission rate depends on an error estimate for a theorem of Schanuel applied to the function field of an asymptotically optimal curve. On the computational side, actual use of the codes will hinge on the solution of new problems in the computational algebraic geometry of curves.

Chapter 15 - Algebraic Methods for Computer Aided Geometric Design

An [$$n, k, d$$n,k,d] code is a binary linear code of block length $$n$$n, dimension $$k$$k and minimum Hamming distance $$d$$d. Since the minimum distance determines the error detection or correction capability, it is desired that $$d$$d is as large as possible for the given block length $$n$$n and dimension $$k$$k. One of the most fundamental problems in coding theory is to construct codes with best possible minimum distances. This problem is very difficult in both theory and practice. During the last decades, it has proved that the class of quasi-cyclic (QC) codes contain many such codes. In this paper, augmentation of binary QC codes is studied. A new augmentation algorithm is presented, and 10 new $$h$$h-generator QC codes that are better than previously known code have been constructed. Furthermore, Construction X has been applied to obtain another 18 new improved binary linear codes. With the standard construction techniques, a total of 124 new binary linear codes that improve the lower bound on the minimum distance have been obtained.

It has been well-known that the class of quasi-cyclic (QC) codes contain many good codes. In this paper, a method to conduct a computer search for binary $2$-generator QC codes is presented, and a large number of good $2$-generator QC codes have been obtained. $5$ new binary QC codes that improve the lower bounds on minimum distance are presented. Furthermore, with new $2$-generator QC codes and Construction X, $2$ new improved binary linear codes are obtained. With the standard construction techniques, another $16$ new binary linear codes that improve the lower bound on the minimum distance have also been obtained.

Low-density parity-check (LDPC) codes constructed by a sparse parity-check matrix are of very fast encoding and decoding algorithms. Another kind of codes, which improved the well-known Gilbert-Varshamov bound, are algebraic geometry codes (Goppa geometry codes) from algebraic curves over finite fields. In the note, we analyze their characteristic of the two class of codes and show that the algebraic geometric codes are seldom LDPC codes.

Nonlinear codes from points of bounded height

Analysis of error correction performance for error correcting codes is very important when using such codes in digital communication systems. At medium-to-high signal-to-noise ratios, the distance spectrum of the error correcting code represents a good indicator for the error correction performance of the code. It is desired that the minimum distance of the code is as large as possible and that the corresponding multiplicity (i.e. the number of codewords having the weight equal to the minimum distance) is as small as possible. If we know an upper bound of the minimum distance of the code, then we have a good indication about the capabilities and the limitations of the code. One of the classes of the error correcting codes with the best performance is that of turbo codes. For such codes, establishing upper bounds on the minimum distance is challenging because it depends on the interleaver component of the turbo code. In this paper we consider turbo codes with component convolutional codes as in the Long Term Evolution standard. The interleaver lengths are of the form $$16 \varPsi $$ or $$48 \varPsi $$ , with $$ \varPsi $$ a product of different prime numbers greater than three. The first achievement in the paper is that for these interleaver lengths, we show that cubic permutation polynomials (CPP), with some constraints on the coefficients, when $$3 \not \mid (p_i-1)$$ for a prime $$p_i > 3$$ , always have a true inverse CPP. The most accurate upper bounds on the minimum distance for turbo codes are achieved by identifying bit information sequences leading to a certain weight of the corresponding turbo-codeword. In this paper we have indentified such bit information sequences by means of the full range dual impulse method to estimate the weight of the turbo-codewords. For the previously mentioned turbo codes and CPP interleavers, we show that the minimum distance is upper bounded by the values of 38, 36, and 28, for three different classes of coefficients. Previously, it was shown that for the same interleaver lengths and for quadratic PP (QPP) interleavers, the upper bound of the minimum distance is equal to 38. Several examples show that $$d_{min}$$ -optimal CPP interleavers are better than $$d_{min}$$ -optimal QPP interleavers because the multiplicities corresponding to the minimum distances for CPPs are about a half of those for QPPs. A theoretical explanation in terms of nonlinearity degrees for this result is given for all considered interleaver lengths and for the class of CPPs for which the upper bound is equal to 38.

${\cal U}$ntil now the representation (i.e. plotting) of curve in Parallel Coordinates is constructed from the point $\leftrightarrow$ line duality. The result is a ``line-curve'' which is seen as the envelope of it's tangents. Usually this gives an unclear image and is at the heart of the ``over-plotting'' problem; a barrier in the effective use of Parallel Coordinates. This problem is overcome by a transformation which provides directly the ``point-curve'' representation of a curve. Earlier this was applied to conics and their generalizations. Here the representation, also called dual, is extended to all planar algebraic curves. Specifically, it is shown that the dual of an algebraic curve of degree $n$ is an algebraic of degree at most $n(n - 1)$ in the absence of singular points. The result that conics map into conics follows as an easy special case. An algorithm, based on algebraic geometry using resultants and homogeneous polynomials, is obtained which constructs the dual image of the curve. This approach has potential generalizations to multi-dimensional algebraic surfaces and their approximation. The ``trade-off'' price then for obtaining {\em planar} representation of multidimensional algebraic curves and hyper-surfaces is the higher degree of the image's boundary which is also an algebraic curve in $\|$-coords.

Coding theoretic and complexity theoretic considerations naturally lead to the question of generating symmetric, sparse, redundant linear systems. This paper provides new way of constructions with better parameters and new lower bounds. Low Density Parity Check (LDPC) codes are linear codes defined by short constraints (a property essential for local testing of a code). Some of the best (theoretically and practically) used codes are LDPC. Symmetric codes are those in which all coordinates look the same, namely there is some transitive group acting on the coordinates which preserves the code. Some of the most commonly used locally testable codes (especially in PCPs and other proof systems), including all low-degree codes, are symmetric. Requiring that a symmetric binary code of length n has large (linear or near-linear) distance seems to suggest a conflict between 1/rate and density (constraint length). In known constructions, if one is constant then the other is almost worst possible - n/poly(log n). Our main positive result simultaneously achieves symmetric low density, constant rate codes generated by a single constraint. We present an explicit construction of a symmetric and transitive binary code of length n, near-linear distance n/(log log n)2, of constant rate and with constraints of length (log n)4. The construction is in the spirit of Tanner codes, namely the codewords are indexed by the edges of a sparse regular expander graph. The main novelty is in our construction of a transitive (non Abelian!) group acting on these edges which preserves the code. Our construction is one instantiation of a framework we call Cayley Codes developed here, that may be viewed as extending zig-zag product to symmetric codes. Our main negative result is that the parameters obtained above cannot be significantly improved, as long as the acting group is solvable (like the one we use). More specifically, we show that in constant rate and linear distance codes (aka good codes) invariant under solvable groups, the density (length of generating constraints) cannot go down to a constant, and is bounded below by log(ω(l)) n if the group has a derived series of length l This negative result precludes natural local tests with constantly many queries for such solvable good codes.

Linear error-correcting codes form the mathematical backbone of modern digital communication and storage systems, but identifying champion linear codes (linear codes achieving or exceeding the best known minimum Hamming distance) remains challenging. By training a transformer to predict the minimum Hamming distance of a class of linear codes and pairing it with a genetic algorithm over the search space, we develop a novel method for discovering champion codes. This model effectively reduces the search space of linear codes needed to achieve champion codes. Our results present the use of this method in the study and construction of error-correcting codes, applicable to codes such as generalised toric, Reed–Muller, Bose–Chaudhuri–Hocquenghem, algebrogeometric, and potentially quantum codes.

This survey is modeled on the CIMPA introductory lectures I gave in Guanajuato in February 2016 about moduli of curves. Surveys on this topic abound and for this reason I decided to focus on few specific aspects giving some illustration of the relation between local and global properties of moduli. The central theme is the interplay between the various notions of “generality” for a curve of genus g. Of course these notes reflect my own view of the subject. For a recent comprehensive text on moduli of curves I refer to Arbarello et al. (Geometry of algebraic curves, Springer, Berlin, 2011).