Showing papers in "Designs, Codes and Cryptography in 1997"
TL;DR: Two general k out of n constructions that are related to those of maximum size arcs or MDS codes and the notion of coloured visual secret sharing schemes is introduced and a general construction is given.
Abstract: The idea of visual k out of n secret sharing schemes was introduced in Naor. Explicit constructions for k = 2 and k = n can be found there. For general k out of n schemes bounds have been described.
Here, two general k out of n constructions are presented. Their parameters are related to those of maximum size arcs or MDS codes. Further, results on the structure of k out of n schemes, such as bounds on their parameters, are obtained. Finally, the notion of coloured visual secret sharing schemes is introduced and a general construction is given.
349 citations
TL;DR: Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes, and which cannot be obtained by derivation or lifting.
Abstract: Planar functions were introduced by Dembowski and Ostrom [4] to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar functions define at least one such affine plane of order 3 ^e for every e ≥ 4 and their projective closures are of Lenz-Barlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner.
304 citations
TL;DR: It is proved that for any integer d there exists a d-regular graph for which any secret sharing scheme has information rate upper bounded by 2/(d+1), which improves on van Dijk's result dik and matches the corresponding lower bound proved by Stinson in [22].
Abstract: A secret sharing scheme is a protocol by means of which a dealer distributes a secret s among a set of participants P in such a way that only qualified subsets of P can reconstruct the value of s whereas any other subset of P, non-qualified to know s, cannot determine anything about the value of the secret.
In this paper we provide a general technique to prove upper bounds on the information rate of secret sharing schemes. The information rate is the ratio between the size of the secret and the size of the largest share given to any participant. Most of the recent upper bounds on the information rate obtained in the literature can be seen as corollaries of our result. Moreover, we prove that for any integer d there exists a d-regular graph for which any secret sharing scheme has information rate upper bounded by 2/(d+1). This improves on van Dijk‘s result dik and matches the corresponding lower bound proved by Stinson in [22].
134 citations
TL;DR: The paper proves that G must be a 2-group and extends previous work to the case that n is a square and proves that H affords a symmetric incidence matrix for the plane.
Abstract: The paper studies a generalized Hadamard matrix H = (g_{ij}) of order n with entries g_{ij} from a group G of order n. We assume that H satisfies: (i) For m
eq k, G = \{g_{mi} g_{ki}^{-1}\mid i = 1, \ldots , n\}; (ii) g_{1i} = g_{i1} = 1 for each i; (iii) g_{ij}^{-1} = g_{ji} for all i, j. Conditions (i) and (ii) occur whenever G is a(P, L) -transitivity for a projective plane of order n. Condition (iii) holds in the case that H affords a symmetric incidence matrix for the plane. The paper proves that G must be a 2-group and extends previous work to the case that n is a square.
76 citations
TL;DR: A condition on the minimum distance of a code to guarantee that the orthogonal code is an almost MDS code is presented, and Evaluation of the MacWilliams identities leads to a closed formula for the weight distribution which turns out to be completely determined for almost M DS codes up to one parameter.
Abstract: The parameters of a linear codeC over GF(q) are given by [n,k,d], where n denotes the length, k the dimension and d the minimum distance of C. The code C is called MDS, or maximum distance separable, if the minimum distance d meets the Singleton bound, i.e. d = n-k+1. Unfortunately, the parameters of an MDS code are severely limited by the size of the field. Thus we look for codes which have minimum distance close to the Singleton bound. Of particular interest is the class of almost MDS codes, i.e. codes for which d=n-k. We will present a condition on the minimum distance of a code to guarantee that the orthogonal code is an almost MDS code. This extends a result of Dodunekov and Landgev Dodunekov. Evaluation of the MacWilliams identities leads to a closed formula for the weight distribution which turns out to be completely determined for almost MDS codes up to one parameter. As a consequence we obtain surprising combinatorial relations in such codes. This leads, among other things, to an answer to a question of Assmus and Mattson 5 on the existence of self-dual [2d,d,d] -codes which have no code words of weight d+1 . Actually there are more codes than Assmus and Mattson expected, but the examples which we know are related to the expected ones.
67 citations
TL;DR: A generalization of the construction gives rise to several new ternary linear codes of dimension six and a 1-1 correspondence between projective linear codes and 2-weight linear codes.
Abstract: We show how to get a 1-1 correspondence between projective linear codes and 2-weight linear codes. A generalization of the construction gives rise to several new ternary linear codes of dimension six.
53 citations
TL;DR: The vector space construction due to Brickell is generalized, and it turns out that the approach of minimal codewords by Massey is a special case of this construction.
Abstract: In this paper, we will generalize the vector space construction due to Brickell. This generalization, introduced by Bertilsson, leads to secret sharing schemes with rational information rates in which the secret can be computed efficiently by each qualified group. A one to one correspondence between the generalized construction and linear block codes is stated, and a matrix characterization of the generalized construction is presented. It turns out that the approach of minimal codewords by Massey is a special case of this construction. For general access structures we present an outline of an algorithm for determining whether a rational number can be realized as information rate by means of the generalized vector space construction. If so, the algorithm produces a secret sharing scheme with this information rate.
39 citations
TL;DR: Upper and lower bounds on the randomness required by the dealer to set up a secret sharing scheme for infinite classes of access structures are provided and a general result on the Randomness of a scheme for the cycle Cn is proved.
Abstract: In this paper we provide upper and lower bounds on the randomness required by the dealer to set up a secret sharing scheme for infinite classes of access structures. Lower bounds are obtained using entropy arguments. Upper bounds derive from a decomposition construction based on combinatorial designs (in particular, t-(v,k,λ) designs). We prove a general result on the randomness needed to construct a scheme for the cycle Cn; when n is odd our bound is tight. We study the access structures on at most four participants and the connected graphs on five vertices, obtaining exact values for the randomness for all them. Also, we analyze the number of random bits required to construct anonymous threshold schemes, giving upper bounds. (Informally, anonymous threshold schemes are schemes in which the secret can be reconstructed without knowledge of which participants hold which shares.)
38 citations
TL;DR: This paper deals with existence for pairwise balanced designs with block sizes 5,6 and 7, block sizes 6,7 and 8 and block sizes 7,8 and 9 and some consequences of these results.
Abstract: This paper deals with existence for pairwise balanced designs with block sizes 5,6 and 7, block sizes 6,7 and 8 and block sizes 7,8 and 9 and some consequences of these results.
37 citations
TL;DR: A classification is given of some optimal ternary linear codes of small length of up to minimum distance 12 for higher dimension where possible.
Abstract: A classification is given of some optimal ternary linear codes of small length. Dimension 2 is classified for every minimum distance. Dimension 3, 4 and 5 is classified up to minimum distance 12. For higher dimension a classification is given where possible.
35 citations
TL;DR: It is shown in particular how codes derived from Artin-Schreier curves, Hermitian curves and Suzuki curves yield classes of universal hash functions which are substantially better than those known before.
Abstract: We describe a new application of algebraic coding theory to universal hashing and authentication without secrecy. This permits to make use of the hitherto sharpest weapon of coding theory, the construction of codes from algebraic curves. We show in particular how codes derived from Artin-Schreier curves, Hermitian curves and Suzuki curves yield classes of universal hash functions which are substantially better than those known before.
TL;DR: Methods to design binary self-dual codes with an automorphism of order two without fixed points are presented and new extremal self- dual codes with previously not known weight enumerators are constructed.
Abstract: Methods to design binary self-dual codes with an automorphism of order two without fixed points are presented. New extremal self-dual [40,20,8], [42,21,8],[44,22,8] and [64,32,12] codes with previously not known weight enumerators are constructed.
TL;DR: Cross-correlation functions are determined for a large class of geometric sequences based on m-sequences in odd characteristic, showing that geometric sequences are candidates for use in spread-spectrum communications systems in which cryptographic security is a factor.
Abstract: Cross-correlation functions are determined for a large class of geometric sequences based on m-sequences in odd characteristic. These sequences are shown to have low cross-correlation values in certain cases. They also have significantly higher linear spans than previously studied geometric sequences. These results show that geometric sequences are candidates for use in spread-spectrum communications systems in which cryptographic security is a factor.
TL;DR: An approximate probability distribution for the maximum order complexity of a random binary sequence is given that enables the development of statistical tests based onmaximum order complexity for the testing of a binary sequence generator.
Abstract: In this paper we give an approximate probability distribution for the maximum order complexity of a random binary sequence This enables the development of statistical tests based on maximum order complexity for the testing of a binary sequence generator These tests are analogous to those based on linear complexity
TL;DR: This paper constructs a nonlinear (64, 237, 12) code as the binary image of an extended cyclic code defined over the integers modulo 4 using Galois rings, and the new code is better than any linear code that is presently known.
Abstract: Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock and Preparata codes, which exist for all lengths 4^m ≥ 16. At length 16 they coincide to give the Nordstrom-Robinson code. This paper constructs a nonlinear (64, 2^37, 12) code as the binary image, under the Gray map, of an extended cyclic code defined over the integers modulo 4 using Galois rings. The Nordstrom-Robinson code is defined in this same way, and like the Nordstrom-Robinson code, the new code is better than any linear code that is presently known.
TL;DR: All extremal double circulant formally self-dual even codes which are not self- dual are classified and the existence of near-extremal formallySelf-duAL even codes are investigated.
Abstract: Formally self-dual even codes have recently been studied. Double circulant even codes are a family of such codes and almost all known extremal formally self-dual even codes are of this form. In this paper, we classify all extremal double circulant formally self-dual even codes which are not self-dual. We also investigate the existence of near-extremal formally self-dual even codes.
TL;DR: It is shown that the weight enumerator of a bordered double circulant self-dual code can be obtained from those of a pure double circular code and its shadow through a relationship between bordered and pure doublecirculant codes.
Abstract: In this paper it is shown that the weight enumerator of a bordered double circulant self-dual code can be obtained from those of a pure double circulant self-dual code and its shadow through a relationship between bordered and pure double circulant codes. As applications, a restriction on the weight enumerators of some extremal double circulant codes is determined and a uniqueness proof of extremal double circulant self-dual codes of length 46 is given. New extremal singly-even [44,22,8] double circulant codes are constructed. These codes have weight enumerators for which extremal codes were not previously known to exist.
TL;DR: The basic necessary condition for the existence of a TD(5, λ; v)-TD(5; u), namely v ≥ 4u, is shown to be sufficient for any λ ≥ 1, except when (v, u) = (6, 1) and λ = 1.
Abstract: The basic necessary condition for the existence of a TD(5, λ; v)-TD(5, λ; u), namely v ≥ 4u, is shown to be sufficient for any λ ≥ 1, except when (v, u) = (6, 1) and λ = 1, and possibly when (v, u) = (10, 1) or (52, 6) and λ = 1. For the case λ = 1, 86 new incomplete transversal designs are constructed. Several construction techniques are developed, and some new incomplete TDs with block size six and seven are also presented.
TL;DR: The main theorem in this paper is that there does not exist an [n,k,d]q code with d = (k-2)qk-1 - ( k-1)qK-2 attaining the Griesmer bound for q ≥ k, k=3,4,5 and for q ≤ 2k-3, k ≥ 6.
Abstract: The main theorem in this paper is that there does not exist an [n,k,d]_q code with d = (k-2)q^{k-1} - (k-1)q^{k-2} attaining the Griesmer bound for q \ge k, ~k=3,4,5k=3,4,5...~ and for q \ge 2k-3, k \ge 6.
TL;DR: A table of upper bounds for K3,2(n1,n2;R), the minimum number of codewords in a covering code with n1 ternary coordinates, n2 binary coordinates, and covering radius R, in the range n = n1 + n2 ≤ 13, R ≤ 3 is presented.
Abstract: A table of upper bounds for K3,2(n1,n2;R), the minimum number of codewords in a covering code with n1 ternary coordinates, n2 binary coordinates, and covering radius R, in the range n = n1 + n2 ≤ 13, R ≤ 3, is presented. Explicit constructions of codes are given to prove the new bounds and verify old bounds. These binary/ternary covering codes can be used as systems for the football pool game. The results include a new binary code with covering radius 1 proving K2(13,1) ≤ 736, and the following upper bound for the football pool problem for 9 matches: K3(9,1) ≤ 1356.
TL;DR: It is shown that with the exception of 66 values of v, this condition is shown to be sufficient for the existence of a resolvable balanced incomplete block design on v points.
Abstract: The necessary condition for the existence of a resolvable balanced incomplete block design on v points, with l = 1 and k = 8, is that v ≡ 8 mod 56. With the exception of 66 values of v, this condition is shown to be sufficient. The largest exceptional value of v is 24480.
TL;DR: This work gives a new condition for realizing a (Ck ⊕ G, Ck × {0}, k, 1)-DF starting from a (G, {0, k,1)-DF and finds new cyclic Steiner 2-designs obtained.
Abstract: Given a subgroup N of an additive group G, a (G,N,k,1) difference family (DF) is a set D of k-subsets of G such that (d − d′ | d, d′ ∈ D, d ≠ d′, D ∈ D) = G − N. Generalizing a construction by Genma, Jimbo, and Mishima [4], we give a new condition for realizing a (Ck ⊕ G, Ck × {0}, k, 1)-DF starting from a (G, {0}, k, 1)-DF. Among the consequences, new cyclic Steiner 2-designs are obtained.
TL;DR: An extension of the applied technique shows that lexicodes overGF(2^{2^k } ) are linear for a wide choice of bases and for a large class of selection criteria, which generalizes a property of Conway and Sloane.
Abstract: Let \V be a list of all words of (GF(2))^n, lexicographically ordered with respect to some basis. Lexicodes are codes constructed from \V by applying a greedy algorithm. A short proof, only based on simple principles from linear algebra, is given for the linearity of these codes. The proof holds for any ordered basis, and for any selection criterion, thus generalizing the results of several authors. An extension of the applied technique shows that lexicodes over GF(2^{2^k}) are linear for a wide choice of bases and for a large class of selection criteria. This result generalizes a property of Conway and Sloane.
TL;DR: In this paper, the problem of finding the maximum number of functions in n variables of which any T form a t-resilient system is investigated, and the problem is reduced to the minimization of the size of certain combinatorial designs, called split orthogonal arrays.
Abstract: A system of (Boolean) functions in n variables is called randomized if the functions preserve the property of their variables to be independent and uniformly distributed random variables. Such a system is referred to as t-resilient if for any substitution of constants for any i variables, where 0 ≤ i ≤ t, the derived system of functions in n-i variables will be also randomized. We investigate the problem of finding the maximum number N(n,t,T) of functions in n variables of which any T form a t-resilient system. This problem is reduced to the minimization of the size of certain combinatorial designs, which we call split orthogonal arrays. We extend some results of design and coding theory, in particular, a duality in bounding the optimal sizes of codes and designs, in order to obtain upper and lower bounds on N(n,t,T). In some cases, these bounds turn out to be very tight. In particular, for some infinite subsequences of integers n they allow us to prove that N(n,3,3)=\frac{2^{n-2}}{n},N(n,3,5)=\sqrt{\frac{2^{n-1}}n} ,N(n,3,\frac n2-1)=n , N(n,\frac n2-1,3)=n, N(n,\frac n2-1,5)=\sqrt{2n}. We also find a connection of the problem considered with the construction of unequal-error-protection codes and superimposed codes for multiple access in the Hamming channel.
TL;DR: It is proved that any (p, k, 1)-DF (p prime) whose base blocks exactly cover p−1/k(k−1) distinct cosets of the k-th roots of unity (mod p), leads to a Ckp-invariantly resolvable cyclic (kp,k,1)-BBD, and proposed several constructions for DF's having this property.
Abstract: A Steiner 2-design is said to be G-invariantly resolvable if admits an automorphism group G and a resolution invariant under G. Introducing and studying resolvable difference families, we characterize the class of G-invariantly resolvable Steiner 2-designs arising from relative difference families over G. Such designs have been already studied by Genma, Jimbo, and Mishima [13] in the case in which G is cyclic. Developping their results, we prove that any (p, k, 1)-DF (p prime) whose base blocks exactly cover p−1/k(k−1) distinct cosets of the k-th roots of unity (mod p), leads to a Ckp-invariantly resolvable cyclic (kp,k,1)-BBD. This induced us to propose several constructions for DF‘s having this property. In such a way we prove, in particular, the existence of a C5p-invariantly resolvable cyclic (5p, 5, 1)-BBD for each prime p = 20n + 1 < 1.000.
TL;DR: A sporadic non-Rédei Type blocking set of PG(2,7) having minimum cardinality is characterized, and an upper bound for the number of nuclei of sets in PG( 2,q) having less than q+1 points is derived.
Abstract: In this paper we characterize a sporadic non-Redei type blocking set of PG (2,7) having minimum cardinality, and derive an upper bound for the number of nuclei of sets in PG (2,q) having less than q+1 points. Our methods involve polynomials over finite fields, and work mainly for planes of prime order.
TL;DR: It is shown that an algorithm designed to solve the Welch–Berlekamp key equation may also be used to solve a more general problem, which can be regarded as a finite analogue of a generalized rational interpolation problem.
Abstract: We show that an algorithm designed to solve the Welch–Berlekamp key equation may also be used to solve a more general problem, which can be regarded as a finite analogue of a generalized rational interpolation problem. As a consequence, we show that a single algorithm exists which can solve both Berlekamp‘s classical key equation (usually solved by the Berlekamp–Massey algorithm) and the Welch–Berlekamp key equation which arise in the decoding of Reed–Solomon codes.
TL;DR: This paper gives classes of Hadamard matrices including at least 170 new orders 2tp,p ≤ 3999, and illustrates how this method increases the versatility of some recent constructions that use block sequences.
Abstract: We further develop the ideas introduced in [4], giving a fuller description of how to obtain Hadamard matrices from certain weighing matrices. Drawing on sequences with zero autocorrelation and the theory of signed groups, we give classes of Hadamard matrices including at least 170 new orders 2^tp,p \leq 3999 . We also illustrate how this method increases the versatility of some recent constructions that use block sequences.
TL;DR: It is proved that the covering radius of a primitive binary BCH code of length q-1 and designed distance 2t+1, where MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn is exactly 2t-1.
Abstract: It is proved that the covering radius of a primitive binary BCH code of length q-1 and designed distance 2t+1, where \[ q=2^m>[(2t-3)(2t-1)!]^2, \] is exactly 2t-1 (the minimum value possible). The bound for q is significantly lower than the one obtained by O. Moreno and C. J. Moreno [9].
TL;DR: The spectrum of PGBTD(n,3) is determined with a fairly small number of exceptions for n and this result is used to establish the existence of a class of Kirkman squares in diagonal form.
Abstract: A generalized balanced tournament design, GBTD(n, k), defined on a kn-set V, is an arrangement of the blocks of a (kn, k, k − 1)-BIBD defined on V into an n × (kn − 1) array such that (1) every element of V is contained in precisely one cell of each column, and (2) every element of V is contained in at most k cells of each row. Suppose we can partition the columns of a GBTD(n, k) into k + 1 sets B1, B2, …, Bk + 1 where |Bi| = n for i = 1, 2, …, k − 2, |Bi| = n−1 for i = k − 1, k and |Bk+1| = 1 such that (1) every element of V occurs precisely once in each row and column of Bi for i = 1, 2, …, k − 2, and (2) every element of V occurs precisely once in each row and column of Bi ∪ Bk+1 for i = k − 1 and i = k. Then the GBTD(n, k) is called partitioned and we denote the design by PGBTD(n, k). The spectrum of GBTD(n, 3) has been completely determined. In this paper, we determine the spectrum of PGBTD(n,3) with, at present, a fairly small number of exceptions for n. This result is then used to establish the existence of a class of Kirkman squares in diagonal form.