17 research outputs found
Quadratic Residue Code
The algebraic decoding of binary quadratic residue codes can be performed using the Peterson or the Berlekamp-Massey algorithm once certain unknown syndromes are determined or eliminated. The technique of determining unknown syndromes is applied to the nonbinary case to decode the expurgated ternary quadratic residue code of length 23
and C.Padilla. Decoding the (41,20,10) Quadratic Residue Code Beyond its ErrorCorrecting Capability
Abstract An algebraic decoding algorithm for the expurgated quadratic residue code of length 41 is presented. The algorithm is guaranteed to produce the correct error-location polynomial whenever an error pattern of weight up to four occurs. An error pattern of weight five is not correctable if it is equidistant from the all-zero codeword and a codeword of weight ten. If an error pattern of weight five occurs, the algorithm will decide whether it is correctable; in the affirmative case, it will either produce the correct error-location polynomial or declare failure. However, the latter outcome, that is, failure, occurs with very low probability. Mathematics Subject Classification: 94B05, 94B15, 94B3
On the Optimality of Finite Constellations from Algebraic Lattices
A construction technique of finite point constellations in n-dimensional spaces from ideals in rings of algebraic integers is described. An algorithm is presented to find constellations with minimum average energy from a given lattice. For comparison, a numerical table of lattice constellations and group codes is computed for spaces of dimension two, three, and four. © 2001
A family of asymptotically good lattices having a lattice in each dimension
A new constructive family of asymptotically good lattices with respect to sphere packing density is presented. The family has a lattice in every dimension n >= 1. Each lattice is obtained from a conveniently chosen integral ideal in a subfield of the cyclotomic field Q(zeta(q)) where q is the smallest prime congruent to 1 modulo n
An Extension of Craig's Family of Lattices
Let p be a prime, and let zeta(p) be a primitive p-th root of unity. The lattices in Craig's family are (p - 1)-dimensional and are geometrical representations of the integral Z[zeta(p)]-ideals (i), where i is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions p - 1 where 149 (i) (j), where p and q are distinct primes and i and fare positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties
On the Construction of New Toric Quantum Codes and Quantum Burst-Error Correcting Codes
A toric quantum error-correcting code construction procedure is presented in
this work. A new class of an infinite family of toric quantum codes is provided
by constructing a classical cyclic code on the square lattice
for all odd integers and,
consequently, new toric quantum codes are constructed on such square lattices
regardless of whether can be represented as a sum of two squares.
Furthermore this work supplies for each the polyomino shapes that
tessellate the corresponding square lattices and, consequently, tile the
lattice . The channel without memory to be considered for these
constructed toric quantum codes is symmetric, since the
-lattice is autodual. Moreover, we propose a quantum
interleaving technique by using the constructed toric quantum codes which shows
that the code rate and the coding gain of the interleaved toric quantum codes
are better than the code rate and the coding gain of Kitaev's toric quantum
codes for , where , and of an infinite class of Bombin and
Martin-Delgado's toric quantum codes. In addition to the proposed quantum
interleaving technique improves such parameters, it can be used for burst-error
correction in errors which are located, quantum data stored and quantum
channels with memory.Comment: Submitted to "Journal of Algebra, Combinatorics, Discrete Structures
and Applications