A µ-[n × n, k] array code C over a field F is a k-dimensional linear space of n × n matrices over F such that every nonzero matrix in C has rank ≥ µ. It is first shown that the dimension of such array codes must satisfy the Singleton-like bound k ≤ n(n−µ+1). A family of so-called maximum-rank µ-[n × n, k = n(n − µ + 1)] array codes is then constructed over every finite field F and for every n and µ, 1 ≤ µ ≤ n. A decoding algorithm is presented for retrieving every Γ ∈ C, given a "received" array Γ + E, where rank(E) = t ≤ (µ − 1)/2. Maximum-rank array codes can be used for decoding crisscross errors in n × n bit arrays, where the erroneous bits are confined to a number t of rows or columns (or both). Our construction proves to be optimal also for this model of errors, which can be found in a number of applications, such as memory chip arrays or magnetic tape recording. Finally, it is shown that the behavior of linear spaces of matrices is quite unique compared with the more general case of linear spaces of n × n × • • • × n hyper-arrays.

Error-correcting codes constitute one of the key ingredients in achieving the high degree of reliability required in modern data transmission and storage systems. This 2006 book introduces the reader to the theoretical foundations of error-correcting codes, with an emphasis on Reed-Solomon codes and their derivative codes. After reviewing linear codes and finite fields, the author describes Reed-Solomon codes and various decoding algorithms. Cyclic codes are presented, as are MDS codes, graph codes, and codes in the Lee metric. Concatenated, trellis, and convolutional codes are also discussed in detail. Homework exercises introduce additional concepts such as Reed-Muller codes, and burst error correction. The end-of-chapter notes often deal with algorithmic issues, such as the time complexity of computational problems. While mathematical rigor is maintained, the text is designed to be accessible to a broad readership, including students of computer science, electrical engineering, and mathematics, from senior-undergraduate to graduate level.

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A list decoding algorithm is presented for [ ] Reed-Solomon (RS) codes over GF ( ), which is capable of correcting more than ( ) 2 errors. Based on a previous work of Sudan, an extended key equation (EKE) is derived for RS codes, which reduces to the classical key equation when the number of errors is limited to ( ) 2 . Generalizing Massey's algorithm that finds the shortest recurrence that generates a given sequence, an algorithm is obtained for solving the EKE in time complexity ( ( ) ), where is a design parameter, typically a small constant, which is an upper bound on the size of the list of decoded codewords. (The case = 1 corresponds to classical decoding of up to ( ) 2 errors where the decoding ends with at most one codeword.) This improves on the time complexity ( ) needed for solving the equations of Sudan's algorithm by a naive Gaussian elimination. The polynomials found by solving the EKE are then used for reconstructing the codewords in time complexity (( log ) ( + log )) using root-finders of degree-univariate polynomials.

A new technique, based on the pseudo-random properties of certain graphs, known as expanders, is used to obtain new simple explicit constructions of asymptotically good codes. In one of the constructions (construction C1 below), the expanders are used to enhance Justesen codes by replicating, shuffling and then regrouping the code coordinates. For a

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