The automorphism groups of the generalized quadratic residue codes

We give the full automorphism groups as groups of semiaffine transformations, of the extended generalized quadratic residue codes. We also present a proof of the Gleason-Prange theorem for the extended generalized quadratic residue codes that relies only on their definition and elementary theory of linear characters     

Generalized quadratic residue codes

A simple definition of generalized quadratic residue codes, that is, quadratic residue codes of block lengthp^{m}, is given, and an account of many of their properties is presented.     

Generalized Hamming weights of linear codes

The generalized Hamming weight, d_r(C), of a binary linear code C is the size of the smallest support of any r-dimensional subcode of C. The parameter d_r(C) determines the code's performance on the wire-tap channel of Type II. Bounds on d_r(C), and in some cases exact expressions, are derived. In particular, a generalized Griesmer bound for d_r(C) is presented and examples are given of codes meeting this bound with equality     

Double circulant quadratic residue codes

We give a lower bound for the minimum distance of double circulant binary quadratic residue codes for primes p/spl equiv//spl plusmn/3(mod8). This bound improves on the square root bound obtained by Calderbank and Beenker, using a completely different technique. The key to our estimates is to apply a result by Helleseth, to which we give a new and shorter proof. Combining this result with the Weil bound leads to the improvement of the Calderbank and Beenker bound. For large primes p, their bound is of order /spl radic/(2p) while our new improved bound is of order 2/spl radic/p. The results can be extended to any prime power q and the modifications of the proofs are briefly indicated.     

Two quaternary quadratic residue codes

The weight distributions of the (13, 6) and the (17, 8) quaternary quadratic residue codes are computed.     

The weight distributions of some binary quadratic residue codes

The weight distributions of binary quadratic residue codes C can be computed from the weight distribution of a subset of C containing one-fourth (resp., one-eighth) of the codewords in C when the length of the code is congruent to 1 (resp., -1) modulo 8. An algorithm to determine the weight distributions of binary cyclic codes is given. As a consequence, the weight distributions of (73,37,13), (89,45,17), and (97,49,15) quadratic residue codes are determined precisely.     

Generalized Hamming weights for linear codes

Motivated by cryptographical applications, the algebraic structure, of linear codes from a new perspective is studied. By viewing the minimum Hamming weight as a certain minimum property of one-dimensional subcodes, a generalized notion of higher-dimensional Hamming weights is obtained. These weights characterize the code performance on the wire-tap channel of type II. Basic properties of generalized weights are derived, the values of these weights for well-known classes of codes are determined, and lower bounds on code parameters are obtained. Several open problems are also listed     

Generalized Hamming weights of q-ary Reed-Muller codes

The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as well as the q-ary Reed-Muller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain condition     

A non-coherent SAC-OCDMA system using extended quadratic congruence codes for two-code keying scheme in passive optical networks

In this paper, we propose a family of extended quadratic congruence codes for two-code keying (TCK) with the corresponding encoding/decoding architecture for passive optical networks (PONs) in spectral amplitude coding optical code division multiple access (OCDMA) systems. The proposed system can simultaneously eliminate multi-user interference (MUI) and further suppress phase-induced intensity noise (PIIN). We reduce the complexity of the encoding/decoding architecture of the optical line terminal reduced by exploiting arrayed waveguide gratings (AWGs) and the properties of the extended quadratic congruence codes (EQC codes). Moreover, we also design a deployment method to increase the number of simultaneous users. Our numerical results demonstrate that the proposed system outperforms the improved quadratic congruence codes (improved QC codes).     

Gray codes and the symbolic dynamics of quadratic maps

Alvarez et al. [1998] used Gray codes to study the symbolic sequences with two symbols which arise from superstable orbits of iterations of quadratic maps. They state several results without proof. A simplified version of this idea is presented, and proofs for the assertions proposed are presented     

Gray codes and 1D quadratic maps

The authors show how the symbolic sequences of same period superstable orbits in 1D quadratic maps are ordered according to Gray codes. Next, the Gray ordering number is introduced, in the interval (0, 1), allowing the simultaneous ordering of symbolic sequences of different period superstable orbits. Likewise, it is shown that Gray ordering number manipulation can determine whether or not a given symbolic sequence exists     

New classes of cyclic extended Goppa codes

We give a condition for a “classical” Goppa (1970) code to have a cyclic extension. This condition follows from the properties of the semilinear automorphism group of the corresponding generalized Reed-Solomon codes. Using this condition, we can construct not only all the known Goppa codes with a cyclic extension, but also new families     

The extended irregular /spl pi/-rotation LDPC codes

We extend the class of deterministic /spl pi/-rotation low-density parity-check codes to a more flexible irregular pattern. The extended codes maintain the advantages of the classic /spl pi/-rotation pattern with easy implementation based on a single vector while providing superb performance.     

Lattice constellations and codes from quadratic number fields

We propose new classes of linear codes over integer rings of quadratic extensions of Q, the field of rational numbers. The codes are considered with respect to a Mannheim metric, which is a Manhattan metric module a two-dimensional (2-D) grid, in particular, codes over Gaussian integers and Eisenstein-Jacobi integers are extensively studied. Decoding algorithms are proposed for these codes when up to two coordinates of a transmitted code vector are affected by errors of arbitrary Mannheim weight. Moreover, we show that the proposed codes are maximum-distance separable (MDS), with respect to the Hamming distance. The practical interest in such Mannheim-metric codes is their use in coded modulation schemes based on quadrature amplitude modulation (QAM)-type constellations, for which neither the Hamming nor the Lee metric is appropriate     

A performance comparison of the binary quadratic residue codes withthe 1/2-rate convolutional codes

The 1/2-rate binary quadratic residue (QR) codes, using binary phase-shift keyed (BPSK) modulation and hard decoding, are presented as an efficient system for reliable communication. Performance results of error correction are obtained both theoretically and by means of computer calculations for a number of binary QR codes. These results are compared with the commonly used 1/2-rate convolutional codes with constraint lengths from 3 to 7 for the hard-decision case. The binary QR codes of different lengths are shown to be equivalent in error-correction performance to some 1/2-rate convolutional codes, each of which has a constraint length K that corresponds to the error-control rate d/n and the minimum distance d of the QR codes     

New optimal partial unit memory codes based on extended BCH codes

Some new optimal binary partial unit memory codes are presented. The codes are constructed from extended BCH codes. A general method for constructing PUM codes based on extended RS and BCH codes is described.<>     

A note on extended quadratic residue codes overGF(9)and their ternary images (Corresp.)

We discuss[2(p + 1), p + 1]double circulant codes which are the ternary images of the[p + 1,(p + 1)/2]extended quadratic residue codes over GF(9). Herepis a prime of the formp = 12k pm 5. As a special result we obtain a[64, 32,18]ternary self-dual code which is the largest known code meeting the bound of Mallows and Sloane.     

Binary codes with improved minimum weights (Corresp.)

A recent table of Helgert and Stinaff gives bounds ford_{max}(n,k), the maximum minimum distance over all binary linear(n,k)error-correcting codes,1 leq k leq n leq 127. Twelve new codes are constructed which improve lower bounds in the table. Two methods are employed: the algebraic puncturing technique of Solomon and Stiffler and generation by combinatorial incidence matrices.     

Turbo codes extended with outer BCH code

The `error floor' observed in several simulations with the turbo codes is verified by calculation of an upper bound to the bit error rate for the ensemble of all interleavers. Also an easy way to calculate the weight enumerator used in this bound is presented. An extended coding scheme is proposed including an outer BCH code correcting a few bit errors     

The extended quadratic residue code is the only (48,24,12) self-dual doubly-even code

An extremal self-dual doubly-even binary (n,k,d) code has a minimum weight d=4/spl lfloor/n/24/spl rfloor/+4. Of such codes with length divisible by 24, the Golay code is the only (24,12,8) code, the extended quadratic residue code is the only known (48,24,12) code, and there is no known (72,36,16) code. One may partition the search for a (48,24,12) self-dual doubly-even code into three cases. A previous search assuming one of the cases found only the extended quadratic residue code. We examine the remaining two cases. Separate searches assuming each of the remaining cases found no codes and thus the extended quadratic residue code is the only doubly-even self-dual (48,24,12) code.