Douglas A. Leonard

Fast erasure and error decoding of algebraic geometry codes up to the Feng-Rao bound

This article gives an errata (that is erasure- and error-) decoding algorithm of one-point algebraic-geometry codes up to the Feng-Rao (1994) designed minimum distance using Sakatas (see Proc. 1995 IEEE Int. Symp. Information Theory, Whistler, BC, Canada, 1995) multidimensional generalization of the Berlekamp-Massey (1969) algorithm and the voting procedure of Feng and Rao.

A generalized Forney formula for algebraic-geometric codes

This correspondence contains a straightforward generalization of decoding of BCH codes to the decoding of algebraic-geometric codes, couched in terms of varieties, ideals, and Grobner bases. This consists of 1) a Berlekamp-Massey-type lattice-shifting row-reduction algorithm with majority voting similar to algorithms in the current literature, 2) a realization that it produces a minimal Grobner basis B for the error-locator ideal I(V) relative to a particular weighted total degree monomial ordering, 3) a factoring of that basis into several minimal PLEX bases, that facilitates finding the variety V of error positions, and 4) a direct generalization of Forneys formula to calculate error magnitudes using functions /spl sigma/p, which are by-products of this factoring.

Finding the defining functions for one-point algebraic-geometry codes

An iterative algorithm is given for producing the parity-check functions for one-point algebraic-geometry (AG) codes, in particular (but not limited to) those related to the two towers of function fields introduced by Garcia and Stichtenoth (1995). This simple method, based on the linearity of the Qth-power map over F/sub Q/, is an alternative to the method of using Laurent series expansions. Examples are given for small values of m with characteristic p=2 for both towers.

Error-locator ideals for algebraic-geometric codes

The error locations for an algebraic-geometric code C*(D,mP) are exactly the common zeros (that is, a projective variety V(I)) of a set (ideal) I of error-locator functions. The paper gives a one-dimensional Berlekamp-Massey version of the Feng-Rao (1993) algorithm for decoding algebraic-geometric codes C*(D,mP). This produces a generating set for I (as an ideal) of size at most /spl rho/ (the smallest positive pole order at P of any function in L(mP)) relative to any error of weight at most e >

The girth of a directed distance-regular graph

Abstract Let G be a distance-regular digraph of short type, that is, with diameter d equal to g − 1 where g is the girth. Suppose the valency k > 1, so that G is not a directed cycle. Then the structure of Γ 1 ( u ) forces g ≤ 8.

Linear cyclic codes of wordlength v over GF(q s ) which are also linear cyclic codes of wordlength sv over GF(q)

Any nonsingular linear transformation φ: GF(qs) → GF(qs) can be used to treat a linear cyclic code ρ of wordlength v over GF(qs) as a linear code φ(ρ) of Wordlength sv over GF(q). This paper determines those linear cyclic codes ρ and transformations φ for which the resulting linear code φ(ρ) is also cyclic.

Efficient Forney functions for decoding AG codes

Using a Forney formula to solve for the error magnitudes in decoding algebraic-geometric (AG) codes requires producing functions /spl sigma//sub P/, which are 0 at all but one point P of the variety of the error-locator ideal. The best such function is produced here in a reasonably efficient way from a lex Grobner basis. This lex basis is, in turn, produced efficiently from a weighted grevlex basis by using the FGLM algorithm. These two steps essentially complete the efficient decoding scheme based on a Forney formula started in the authors previous work (see ibid., vol.42, p.1263-8, 1996).

Directed distance regular graphs with the Q -polynomial property

Abstract This paper shows that the parameters of a non-symmetric, metric, cometric association scheme are determined by the eigenvalues and dual eigenvalues of the scheme. All the work is done at the level of the associated dual pair of algebras.

A weighted module view of integral closures of affine domains of type I

A type I presentation

Non-symmetric, metric, cometric association schemes are self-dual

S=R/J