Ron M. Roth

Maximum-rank array codes and their application to crisscross error correction

A mu -(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 >or= mu . It is first shown that the dimension of such array codes must satisfy the Singleton-like bound k >

Efficient decoding of Reed-Solomon codes beyond half the minimum distance

A list decoding algorithm is presented for [n,k] Reed-Solomon (RS) codes over GF(q), which is capable of correcting more than [(n-k)/2] errors. Based on a previous work of Sudan (see J. Compl., vol.13, p.180-93, 1997), 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 [(n-k)/2]. Generalizing Masseys (1969) algorithm that finds the shortest recurrence that generates a given sequence, an algorithm is obtained for solving the EKE in time complexity O(l/spl middot/(n-k)/sup 2/), where l is a design parameter, typically a small constant, which s an upper bound on the size of the list of decoded codewords. (The case l=1 corresponds to classical decoding of up to [(n-k)/2] errors where the decoding ends with at most one codeword.) This improves on the time complexity O(n/sup 3/) needed for solving the equations of Sudans algorithm by a naive Gaussian elimination. The polynomials found by solving the EKE are then used for reconstructing the codewords in time complexity O((llog/sup 2/l)k(n+llogq)) using root-finders of degree-l univariate polynomials.

New array codes for multiple phased burst correction

An optimal family of array codes over GF(q) for correcting multiple phased burst errors and erasures, where each phased burst corresponds to an erroneous or erased column in a code array, is introduced. As for erasures, these array codes have an efficient decoding algorithm which avoids multiplications (or divisions) over extension fields, replacing these operations with cyclic shifts of vectors over GF(q). The erasure decoding algorithm can be adapted easily to handle single column errors as well. The codes are characterized geometrically by means of parity constraints along certain diagonal lines in each code array, thus generalizing a previously known construction for the special case of two erasures. Algebraically, they can be interpreted as Reed-Solomon codes. When q is primitive in GF(q), the resulting codes become (conventional) Reed-Solomon codes of length P over GF(q/sup p-1/), in which case the new erasure decoding technique can be incorporated into the Berlekamp-Massey algorithm, yielding a faster way to compute the values of any prescribed number of errors. >

Lee-metric BCH codes and their application to constrained and partial-response channels

Shows that each code in a certain class of BCH codes over GF(p), specified by a code length n/spl les/p/sup m/-1 and a runlength r/spl les/(p-1)/2 of consecutive roots in GF(p/sup m/), has minimum Lee distance /spl ges/2r. For the very high-rate range these codes approach the sphere-packing bound on the minimum Lee distance. Furthermore, for a given r, the length range of these codes is twice as large as that attainable by Berlekamps (1984) extended negacyclic codes. The authors present an efficient decoding procedure, based on Euclids algorithm, for correcting up to r-1 errors and detecting r errors, that is, up to the number of Lee errors guaranteed by the designed minimum Lee distance 2r. Bounds on the minimum Lee distance for r/spl ges/(p+1)/2 are provided for the Reed-Solomon case, i.e., when the BCH code roots are in GF(p). The authors present two applications. First, Lee-metric BCH codes can be used for protecting against bitshift errors and synchronization errors caused by insertion and/or deletion of zeros in (d, k)-constrained channels. Second, the code construction with its decoding algorithm can be formulated over the integer ring, providing an algebraic approach to correcting errors in partial-response channels where matched spectral-null codes are used. >

On generator matrices of MDS codes (Corresp.)

It is shown that the family of q -ary generalized Reed-Solomon codes is identical to the family of q -ary linear codes generated by matrices of the form [I|A] , where I is the identity matrix, and A is a generalized Cauchy matrix. Using Cauchy matrices, a construction is shown of maximal triangular arrays over GF (q) , which are constant along diagonals in a Hankel matrix fashion, and with the property that every square subarray is a nonsingular matrix. By taking rectangular subarrays of the described triangles, it is possible to construct generator matrices [I|A] of maximum distance separable codes, where A is a Hankel matrix. The parameters of the codes are (n,k,d) , for 1 \leq n \leq q+ 1, 1 \leq k \leq n , and d=n-k+1 .

Efficient coding schemes for the hard-square model

The hard-square model, also known as the two-dimensional (2-D) (1, /spl infin/)-RLL constraint, consists of all binary arrays in which the 1s are isolated both horizontally and vertically. Based on a certain probability measure defined on those arrays, an efficient variable-to-fixed encoder scheme is presented that maps unconstrained binary words into arrays that satisfy the hard-square model. For sufficiently large arrays, the average rate of the encoder approaches a value which is only 0.1% below the capacity of the constraint. A second, fixed-rate encoder is presented whose rate for large arrays is within 1.2% of the capacity value.

Is code equivalence easy to decide

We study the computational difficulty of deciding whether two matrices generate equivalent linear codes, i.e., codes that consist of the same codewords up to a fixed permutation on the codeword coordinates. We call this problem code equivalence. Using techniques from the area of interactive proofs, we show on the one hand, that under the assumption that the polynomial-time hierarchy does not collapse, code equivalence is not NP-complete. On the other hand, we present a polynomial-time reduction from the graph isomorphism problem to code equivalence. Thus if one could find an efficient (i.e., polynomial-time) algorithm for code equivalence, then one could settle the long-standing problem of determining whether there is an efficient algorithm for solving graph isomorphism.

Interpolation and approximation of sparse multivariate polynomials over GF(2)

A function

Improved bit-stuffing bounds on two-dimensional constraints

f:\{ 0,1\} ^n \to \{ 0,1\}

On cyclic MDS codes of length q over GF(q) (Corresp.)

is called t-sparse if the n-variable polynomial representation of f over