Lexicographic Codes: Constructions Bounds, and Trellis Complexity

IEEE Transactions on Information Theory, 1986

FELLOW, IEEE A~.~hslruct-Lexicographic codes, or lexicodes, are defined by various versions of the greedy algorithm. The theory of these codes is closely related to the theory of certain impartial games, which leads to a number of surprising properties. For example, lexicodes over an alphabet of size B = 2" are closed under addition;' while if B = 22u the lexicodes are closed under multiplication by scalars, where addition and multiplication are in the nim sense explained in the text. Hamming codes and the binary Colay codes are lexicodes. Remarkably simple constructions are given for the Steiner systems S(5,6,12) and S(5,8,24). Several record-breaking constant weight codes are also constructed.

Designs, Codes and Cryptography, 1999

An ordered list of binary words of length n is called a distance-preserving 〈m, n〉-code, if the list distance between two words is equal to their Hamming distance, for distances up to m. A technique for constructing cyclic 〈m, n〉-codes is presented, based on the standard Gray code and on some simple tools from linear algebra.

Discrete Mathematics, 1985

s), s = 2 ~ and n >12, denote the Desarguesian projective space of projective dimension n over the Galois field Fs. The set of its subsets with set theoretic symmetric difference as addition is a vector space over F2. For 1 ~< t n-1, let Ct(n, s) denote its subspace generated by the t-flats of PG(n, s) and for w c_ PG(n, s), let [wl denote the cardinality (or weight) of w. Our object in this note is to present a purely geometric proof of the following theorem proved independently by Smith [5] and Delsarte et al. [2]. Theorem. For s = 2", n > 1 and 0<t<n, the words of Ct(n, s) of least non-zero weight are precisely the t-fiats of PG(n, s). Some crucial parts of the proof are contained in the following lemmas.

Finite Fields and Their Applications, 2017

In this paper we determine many values of the second least weight of codewords, also known as the next-to-minimal Hamming weight, for a type of affine variety codes, obtained by evaluating polynomials of degree up to d on the points of a cartesian product of n subsets of a finite field F q. Such codes firstly appeared in a work by O. Geil and C. Thomsen (see [12]) as a special case of the so-called weighted Reed-Muller codes, and later appeared independently in a work by H. López, C. Rentería-Marquez and R. Villarreal (see [16]) named as affine cartesian codes. Our work extends, to affine cartesian codes, the results obtained by Rolland in [17] for generalized Reed-Muller codes.

Information Theory, IEEE …, 1993

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 39, NO. 1. JANUARY 1993 Proof: This follows immediately from the definition of C(A, I, q, m) in Section II and the definition of algebraic-geometric code Cl(D, mPoo), see [8, p. 55] and [15, p. 266]. □ Proof of ...

IEEE Transactions on Information Theory, 1999

New constructions of linear nonbinary codes with covering radius R = 2 are proposed. They are in part modifications of earlier constructions by the author and in part are new. Using a starting code with R = 2 as a "seed" these constructions yield an infinite family of codes with the same covering radius. New infinite families of codes with R = 2 are obtained for all alphabets of size q 4 and all codimensions r 3 with the help of the constructions described. The parameters obtained are better than those of known codes. New estimates for some partition parameters in earlier known constructions are used to design new code families. Complete caps and other saturated sets of points in projective geometry are applied as starting codes. A table of new upper bounds on the length function for q = 4; 5; 7; R = 2; and r 24 is included.

Theoretical Computer Science, 2002

The natural correspondence between preÿx codes and trees is explored, generalizing the results obtained in Giammarresi et al. (Theoret. Comput. Sci. 205 (1998) 1459) for the lattice of ÿnite trees under division and the lattice of ÿnite maximal preÿx codes. Joins and meets of preÿx codes are studied in this light in connection with such concepts as ÿniteness, maximality and varieties of rational languages. Decidability results are obtained for several problems involving rational preÿx codes, including the solution to the primeness problem.

IEEE Transactions on Information Theory, 1995

The (120,6,78) QT code has weight distribution Weight 0 78 81 87 96 +---1 528 80 96 24 Count

IEEE Transactions on Information Theory, 2000

Designs, Codes and Cryptography, 2005

Let V be a list of all vectors of GF(q)n , lexicographically ordered with respect to some basis. Algorithms which search list V from top to bottom, any time selecting a codeword which satisfies some criterion, are called greedy algorithms and the resulting set of codewords is called a lexicode. In many cases such a lexicode turns out to be linear. In this paper we present a greedy algorithm for the construction of a large class of linear q-ary lexicodes which generalizes the algorithms of several other papers and puts these into a wider framework. By applying this new method, one can produce linear lexicodes which cannot be constructed by previous algorithms, because the characteristics or the underlying field of the codes do not meet the conditions of those algorithms.