Almost Affine Codes (original) (raw)
Discrete Mathematics, 1989
Let F be a set of f points in a finite projective geometry PG(t, q) of t dimensions where t 2 2, f 3 1 and q is a prime power. If (a) IF n HI 3 m for any hyperplane H in PG(t, q) and (b) IF fl H( = m for some hyperplane H in PG(t, q), then F is said to be an {f, m; t, q}-minhyper (or an {f, m; t, q}minihyper) where m 30 and IAl denotes the number of points in the set A. The concept of a min . hyper (called a minihyper) has been introduced by Hamada and Tamari [22]. In the special case t = 2, an {f, m; 2, q}-min . hyper F is called an m-blocking set if F contains no l-flat in PG(2, q).
Erasure-Resilient Codes from Affine Spaces
In this paper, we investigate erasure-resilient codes coming from Steiner 2-designs with block size k which can correct up to any k erasures. In view of applications it is desirable that such a code can also correct as many erasures of higher order as possible. Our main result is that the erasure-resilient code constructed from an affine space with block size q – a special Steiner 2-design – can not only correct up to any q erasures but even up to any 2q − 1 erasures except for a small set of so-called bad erasures if q is a power of some odd prime number. This gives a new family of erasure-resilient codes which is asymptotically optimal in view of the check bit overhead.
Group code structures of affine-invariant codes
Journal of Algebra, 2010
A group code structure of a linear code is a description of the code as one-sided or two-sided ideal of a group algebra of a finite group. In these realizations, the group algebra is identified with the ambient space, and the group elements with the coordinates of the ambient space. It is well known that every affine-invariant code of length p m , with p prime, can be realized as an ideal of the group algebra FI, where I is the underlying additive group of the field with p m elements. In this paper we describe all the group code structures of an affine-invariant code of length p m in terms of a family of maps from I to the group of automorphisms of I.
Further improvements on the designed minimum distance of algebraic geometry codes
Journal of Pure and Applied Algebra, 2009
In the literature about algebraic geometry codes one finds a lot of results improving Goppa's minimum distance bound. These improvements often use the idea of "shrinking" or "growing" the defining divisors of the codes under certain technical conditions. The main contribution of this article is to show that most of these improvements can be obtained in a unified way from one (rather simple) theorem. Our result does not only simplify previous results but it also improves them further.
On the order bound of one-point algebraic geometry codes
Journal of Pure and Applied Algebra, 2009
Let S ={si}i∈IN ⊆ IN be a numerical semigroup. For each i ∈ IN, let ν(si) denote the number of pairs (si−sj, sj) ∈ S 2 : it is well-known that there exists an integer m such that the sequence {ν(si)}i∈IN is non-decreasing for i > m. The problem of finding m is solved only in special cases. By way of a suitable parameter t, we improve the known bounds for m and in several cases we determine m explicitely. In particular we give the value of m when the Cohen-Macaulay type of the semigroup is three or when the multiplicity is lower or equal to six. When S is the Weierstrass semigroup of a family {Ci}i∈IN of one-point algebraic geometry codes, these results give better estimates for the order bound on the minimum distance of the codes {Ci}.
A ug 2 02 1 Some hypersurfaces over finite fields , minimal codes and secret sharing schemes
2021
Linear error-correcting codes can be used for constructing secret sharing schemes; however finding in general the access structures of these secret sharing schemes and, in particular, determining efficient access structures is difficult. Here we investigate the properties of certain algebraic hypersurfaces over finite fields, whose intersection numbers with any hyperplane only takes a few values. These varieties give rise to q-divisible linear codes with at most 5 weights. Furthermore, for q odd these codes turn out to be minimal and we characterize the access structures of the secret sharing schemes based on their dual codes. Indeed, the secret sharing schemes thus obtained are democratic that is, each participant belongs to the same number of minimal access sets.
An Assmus–Mattson theorem for codes over commutative association schemes
Designs, Codes and Cryptography, 2017
We prove an Assmus-Mattson-type theorem for block codes where the alphabet is the vertex set of a commutative association scheme (say, with s classes). This in particular generalizes the Assmus-Mattson-type theorems for Z 4-linear codes due to Tanabe (2003) and Shin, Kumar, and Helleseth (2004), as well as the original theorem by Assmus and Mattson (1969). The weights of a code are s-tuples of non-negative integers in this case, and the conditions in our theorem for obtaining t-designs from the code involve concepts from polynomial interpolation in s variables. The Terwilliger algebra is the main tool to establish our results.
There Are Not Non-obvious Cyclic Affine-invariant Codes
Lecture Notes in Computer Science, 2009
We show that an affine-invariant code C of length p m is not permutation equivalent to a cyclic code except in the obvious cases: m = 1 or C is either {0}, the repetition code or its dual.
On the Composition of Secret Sharing Schemes Related to Codes
Discrete Mathematics, Algorithms and Applications, 2014
In this paper we construct a subclass of the composite access structure introduced in [9] based on schemes realizing the structure given by the set of codewords of minimal support of linear codes. This class enlarges the iterated threshold class studied in the same paper. Furthermore all the schemes on this paper are ideal (in fact they allow a vector space construction) and we arrived to give a partial answer to a conjecture stated in . Finally, as a corollary we proof that all the monotone access structures based on all the minimal supports of a code can be realized by a vector space construction.
On the unique representation of very strong algebraic geometry codes
Designs, Codes and Cryptography, 2014
This paper addresses the question of retrieving the triple (X , P, E) from the algebraic geometry code C = C L (X , P, E), where X is an algebraic curve over the finite field F q , P is an n-tuple of F q-rational points on X and E is a divisor on X. If deg(E) ≥ 2g + 1 where g is the genus of X , then there is an embedding of X onto Y in the projective space of the linear series of the divisor E. Moreover, if deg(E) ≥ 2g + 2, then I(Y), the vanishing ideal of Y, is generated by I 2 (Y), the homogeneous elements of degree two in I(Y). If n > 2 deg(E), then I 2 (Y) = I 2 (Q), where Q is the image of P under the map from X to Y. These three results imply that, if 2g + 2 ≤ m < 1 2 n, an AG representation (Y, Q, F) of the code C can be obtained just using a generator