Des Codes Cryptogr Research Articles

Extremal properties of t-SEEDs and recursive constructions

A t-spontaneous emission error design, called t-SEED for short, is a combinatorial design introduced by Beth et al. (Des Codes Cryptogr 29:51---70, 2003) in relation to a quantum jump code. In this article, firstly, it is shown that an optimal t-SEED attaining a given upper bound is a large set of Steiner t-designs. Secondly, we present some recursive constructions of t-SEEDs. Moreover, an application to secret sharing scheme by utilizing the properties of a t-SEED is also discussed.

The geometric dimension of some small configurations

Recently, Jungnickel and Tonchev (Des Codes Cryptogr, doi:10.1007/s10623-012-9636-z, 2012) introduced new invariants for simple incidence structures \({\mathcal{D}}\), which admit both a coding theoretic and a geometric description. Geometrically, one considers embeddings of \({\mathcal{D}}\) into projective geometries \({\Pi} = PG(n, q)\), where an embedding means identifying the points of \({\mathcal{D}}\) with a point set V in \({\Pi}\) in such a way that every block of \({\mathcal{D}}\) is induced as the intersection of V with a suitable subspace of \({\Pi}\). Then the new invariant, the geometric dimension\({\mathrm{geomdim}_{q}\mathcal{D}}\) of \({\mathcal{D}}\), is the smallest value of n for which \({\mathcal{D}}\) may be embedded into the n-dimensional projective geometry PG(n, q). It is the aim of this paper to discuss a few additional general results regarding these invariants, and to determine them for some further examples, mainly some small configurations; this will answer some problems posed in (Des Codes Cryptogr, doi:10.1007/s10623-012-9636-z, 2012).

Inner balance of symmetric designs

A triple array is a row-column design which carries two balanced incomplete block designs (BIBDs) as substructures. McSorley et al. (Des Codes Cryptogr 35: 21---45, 2005), Section 8, gave one example of a triple array that also carries a third BIBD, formed by its row-column intersections. This triple array was said to be balanced for intersection, and they made a search for more such triple arrays among all potential parameter sets up to some limit. No more examples were found, but some candidates with suitable parameters were suggested. We define the notion of an inner design with respect to a block for a symmetric BIBD and present criteria for when this inner design can be balanced. As triple arrays in the canonical case correspond to SBIBDs, this in turn yields new existence criteria for triple arrays balanced for intersection. In particular, we prove that the residual design of the related SBIBD with respect to the defining block must be quasi-symmetric, and give necessary and sufficient conditions on the intersection numbers. This, together with our parameter bounds enable us to exclude the suggested triple array candidates in McSorley et al. (Des Codes Cryptogr 35: 21---45, 2005) and many others in a wide search. Further we investigate the existence of SBIBDs whose inner designs are balanced with respect to every block. We show as a key result that such SBIBDs must possess the quasi-3 property, and we answer the existence question for all known classes of these designs.

On the functional codes defined by quadrics and Hermitian varieties

In recent years, functional codes have received much attention. In his PhD thesis, F.A.B. Edoukou investigated various functional codes linked to quadrics and Hermitian varieties defined in finite projective spaces (Edoukou, PhD Thesis, 2007). This work was continued in (Edoukou et al., Des Codes Cryptogr 56:219---233, 2010; Edoukou et al., J Pure Appl Algebr 214:1729---1739, 2010; Hallez and Storme, Finite Fields Appl 16:27---35, 2010), where the results of the thesis were improved and extended. In particular, Hallez and Storme investigated the functional codes $${C_2(\mathcal{H})}$$ , with $${\mathcal{H}}$$ a non-singular Hermitian variety in PG(N, q 2). The codewords of this code are defined by evaluating the points of $${\mathcal{H}}$$ in the quadratic polynomials defined over $${\mathbb{F}_{q^2}}$$ . We now present the similar results for the functional code $${C_{Herm}(\mathcal{Q})}$$ . The codewords of this code are defined by evaluating the points of a non-singular quadric $${\mathcal{Q}}$$ in PG(N, q 2) in the polynomials defining the Hermitian varieties of PG(N, q 2).

Quadratic functions with prescribed spectra

We study a class of quadratic p-ary functions $${{\mathcal{F}}_{p,n}}$$ from $${\mathbb{F}_{p^n}}$$ to $${\mathbb{F}_p, p \geq 2}$$ , which are well-known to have plateaued Walsh spectrum; i.e., for each $${b \in \mathbb{F}_{p^n}}$$ the Walsh transform $${\hat{f}(b)}$$ satisfies $${|\hat{f}(b)|^2 \in \{ 0, p^{(n+s)}\}}$$ for some integer 0 ? s ? n ? 1. For various types of integers n, we determine possible values of s, construct $${{\mathcal{F}}_{p,n}}$$ with prescribed spectrum, and present enumeration results. Our work generalizes some of the earlier results, in characteristic two, of Khoo et. al. (Des Codes Cryptogr, 38, 279---295, 2006) and Charpin et al. (IEEE Trans Inf Theory 51, 4286---4298, 2005) on semi-bent functions, and of Fitzgerald (Finite Fields Appl 15, 69---81, 2009) on quadratic forms.

Small tight sets of hyperbolic quadrics

An x-tight set of a hyperbolic quadric Q+(2n + 1, q) can be described as a set M of points with the property that the number of points of M in the tangent hyperplanes of points of M is as big as possible. We show that such a set is necessarily the union of x mutually disjoint generators provided that x ≤ q and n ≤ 3, or that x < q, n ≥ 4 and q ≥ 71. This unifies and generalizes many results on x-tight sets that are presently known, see (J Comb Theory Ser A 114(7):1293–1314 [1], J Comb Des 16(4):342–349 [5], Des Codes Cryptogr 50:187–201 [4], Adv Geom 4(3):279–286 [8], Bull Lond Math Soc 42(6):991–996 [11]).

New invariants for incidence structures

We exhibit a new, surprisingly tight, connection between incidence structures, linear codes, and Galois geometry. To this end, we introduce new invariants for finite simple incidence structures \({\mathcal{D}}\), which admit both an algebraic and a geometric description. More precisely, we will associate one invariant for the isomorphism class of \({\mathcal{D}}\) with each prime power q. On the one hand, we consider incidence matrices M with entries from GF(qt) for the complementary incidence structure \({\mathcal{D}^*}\), where t may be any positive integer; the associated codes C = C(M) spanned by M over GF(qt); and the corresponding trace codes Tr(C(M)) over GF(q). The new invariant, namely the q-dimension\({{\rm dim}_q(\mathcal{D}^*)}\) of \({\mathcal{D}^*}\), is defined to be the smallest dimension over all trace codes which may be obtained in this manner. This modifies and generalizes the q-dimension of a design as introduced in Tonchev (Des Codes Cryptogr 17:121–128, 1999). On the other hand, we consider embeddings of \({\mathcal{D}}\) into projective geometries \({\Pi = PG(n, q)}\), where an embedding means identifying the points of \({\mathcal{D}}\) with a point set V in \({\Pi}\) in such a way that every block of \({\mathcal{D}}\) is induced as the intersection of V with a suitable subspace of \({\Pi}\). Our main result shows that the q-dimension of \({\mathcal{D}^*}\) always coincides with the smallest value of N for which \({\mathcal{D}}\) may be embedded into the (N − 1)-dimensional projective geometry PG(N − 1, q). We also give a necessary and sufficient condition when actually an embedding into the affine geometry AG(N − 1, q) is possible. Several examples and applications will be discussed: designs with classical parameters, some Steiner designs, and some configurations.

The complexity of the graph access structures on six participants

In this paper, we deal with determining the exact values for the complexity of the five graph access structures on six participants, which remained as open problems in van Dijk's paper (Des Codes Cryptogr 6:143---169, 1995). We prove that the complexity for each of the five graph access structures is equal to 7/4.

A Hamada type characterization of the classical geometric designs

The dimension of a combinatorial design $${{\mathcal D}}$$ over a finite field F = GF(q) was defined in (Tonchev, Des Codes Cryptogr 17:121---128, 1999) as the minimum dimension of a linear code over F that contains the blocks of $${{\mathcal D}}$$ as supports of nonzero codewords. There it was proved that, for any prime power q and any integer n � 2, the dimension over F of a design $${{\mathcal D}}$$ that has the same parameters as the complement of a classical point-hyperplane design PG n-1(n, q) or AG n-1(n, q) is greater than or equal to n + 1, with equality if and only if $${{\mathcal D}}$$ is isomorphic to the complement of the classical design. It is the aim of the present paper to generalize this Hamada type characterization of the classical point-hyperplane designs in terms of associated codes over F = GF(q) to a characterization of all classical geometric designs PG d (n, q), where 1 � d � n � 1, in terms of associated codes defined over some extension field E = GF(q t ) of F. In the affine case, we conjecture an analogous result and reduce this to a purely geometric conjecture concerning the embedding of simple designs with the parameters of AG d (n, q) into PG(n, q). We settle this problem in the affirmative and thus obtain a Hamada type characterization of AG d (n, q) for d = 1 and for d > (n � 2)/2.

A counterexample concerning the 3-error linear complexity of 2 n -periodic binary sequences

In this article, we present a counterexample to Theorem 4.2 and Theorem 5.2 by Kavuluru (Des Codes Cryptogr 53:75---97, 2009). We conclude that the counting functions for the number of 2 n -periodic binary sequences with fixed 3-error linear complexity by Kavuluru are not correct.

Divisibility of polynomials over finite fields and combinatorial applications

Consider a maximum-length shift-register sequence generated by a primitive polynomial f over a finite field. The set of its subintervals is a linear code whose dual code is formed by all polynomials divisible by f. Since the minimum weight of dual codes is directly related to the strength of the corresponding orthogonal arrays, we can produce orthogonal arrays by studying divisibility of polynomials. Munemasa (Finite Fields Appl 4(3):252---260, 1998) uses trinomials over $${\mathbb{F}_2}$$ to construct orthogonal arrays of guaranteed strength 2 (and almost strength 3). That result was extended by Dewar et al. (Des Codes Cryptogr 45:1---17, 2007) to construct orthogonal arrays of guaranteed strength 3 by considering divisibility of trinomials by pentanomials over $${\mathbb{F}_2}$$ . Here we first simplify the requirement in Munemasa's approach that the characteristic polynomial of the sequence must be primitive: we show that the method applies even to the much broader class of polynomials with no repeated roots. Then we give characterizations of divisibility for binomials and trinomials over $${\mathbb{F}_3}$$ . Some of our results apply to any finite field $${\mathbb{F}_q}$$ with q elements.

On quasi-symmetric designs with intersection difference three

In a recent paper, Pawale (Des Codes Cryptogr, 2010) investigated quasi-symmetric 2-(v, k, ?) designs with intersection numbers x > 0 and y = x + 2 with ? > 1 and showed that under these conditions either ? = x + 1 or ? = x + 2, or $${\mathcal{D}}$$ is a design with parameters given in the form of an explicit table, or the complement of one of these designs. In this paper, quasi-symmetric designs with y ? x = 3 are investigated. It is shown that such a design or its complement has parameter set which is one of finitely many which are listed explicitly or ? ? x + 4 or 0 ? x ? 1 or the pair (?, x) is one of (7, 2), (8, 2), (9, 2), (10, 2), (8, 3), (9, 3), (9, 4) and (10, 5). It is also shown that there are no triangle-free quasi-symmetric designs with positive intersection numbers x and y with y = x + 3.

Permutation polynomials EA-equivalent to the inverse function over GF (2 n )

In this paper, a proof is given that there does not exist a linearized polynomial $L(x)\in\mathbb{F}_{2^n}[x]$ such that x ???1?+?L(x) is a permutation on $\mathbb{F}_{2^n}$ when n???5, which is proposed as a conjecture in Li and Wang (Des Codes Cryptogr 58(3):259---269, 2011). As a consequence of this result, if a permutation is EA-equivalent to the inverse function over $\mathbb{F}_{2^n}$ , then it is affine equivalent to the inverse mapping when n???5.

Quaternary Golay sequence pairs I: even length

The origin of all 4-phase Golay sequences and Golay sequence pairs of even length at most 26 is explained. The principal techniques are the three-stage construction of Fiedler, Jedwab and Parker involving multi-dimensional Golay arrays, and a "sum---difference" construction that modifies a result due to Eliahou, Kervaire and Saffari. The existence of 4-phase seed pairs of lengths 3, 5, 11, and 13 is assumed; their origin is considered in (Gibson and Jedwab, Des Codes Cryptogr, 2010).

Exponential bounds on the number of designs with affine parameters

AbstractIt is well‐known that the number of designs with the parameters of a classical design having as blocks the hyperplanes in PG(n, q) or AG(n, q), n≥3, grows exponentially. This result was extended recently [D. Jungnickel, V. D. Tonchev, Des Codes Cryptogr, published online: 23 May, 2009] to designs having the same parameters as a projective geometry design whose blocks are the d‐subspaces of PG(n, q), for any 2≤d≤n−1. In this paper, exponential lower bounds are proved on the number of non‐isomorphic designs having the same parameters as an affine geometry design whose blocks are the d‐subspaces of AG(n, q), for any 2≤d≤n−1, as well as resolvable designs with these parameters. An exponential lower bound is also proved for the number of non‐isomorphic resolvable 3‐designs with the same parameters as an affine geometry design whose blocks are the d‐subspaces of AG(n, 2), for any 2≤d≤n−1. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 475–487, 2010

A family of non class-regular symmetric transversal designs of spread type

Many classes of symmetric transversal designs have been constructed from generalized Hadamard matrices and they are necessarily class regular. In (Hiramine, Des Codes Cryptogr 56:21---33, 2010) we constructed symmetric transversal designs using spreads of $${\mathbb{Z}_p^{2n}}$$ with p a prime. In this article we show that most of them admit no class regular automorphism groups. This implies that they are never obtained from generalized Hadamard matrices. As far as we know, this is the first infinite family of non class-regular symmetric transversal designs.

On linear sets on a projective line

Linear sets generalise the concept of subgeometries in a projective space. They have many applications in finite geometry. In this paper we address two problems for linear sets: the equivalence problem and the intersection problem. We consider linear sets as quotient geometries and determine the exact conditions for two linear sets to be equivalent. This is then used to determine in which cases all linear sets of rank 3 of the same size on a projective line are (projectively) equivalent. In (Donati and Durante, Des Codes Cryptogr, 46:261---267), the intersection problem for subgeometries of PG(n, q) is solved. The intersection of linear sets is much more difficult. We determine the intersection of a subline PG(1, q) with a linear set in PG(1, q h ) and investigate the existence of irregular sublines, contained in a linear set. We also derive an upper bound, which is sharp for odd q, on the size of the intersection of two different linear sets of rank 3 in PG(1, q h ).

A study of (x(q + 1), x; 2, q)-minihypers

In this paper, we study the weighted (x(q + 1), x; 2, q)-minihypers. These are weighted sets of x(q + 1) points in PG(2, q) intersecting every line in at least x points. We investigate the decomposability of these minihypers, and define a switching construction which associates to an (x(q + 1), x; 2, q)-minihyper, with x ? q 2 ? q, not decomposable in the sum of another minihyper and a line, a (j(q + 1), j; 2, q)-minihyper, where j = q 2 ? q ? x, again not decomposable into the sum of another minihyper and a line. We also characterize particular (x(q + 1), x; 2, q)-minihypers, and give new examples. Additionally, we show that (x(q + 1), x; 2, q)-minihypers can be described as rational sums of lines. In this way, this work continues the research on (x(q + 1), x; 2, q)-minihypers by Hill and Ward (Des Codes Cryptogr 44:169---196, 2007), giving further results on these minihypers.

On orthogonal generalized equitable rectangles

In this note, we give a complete solution of the existence of orthogonal generalized equitable rectangles, which was raised as an open problem in by Stinson (Des Codes Cryptogr 45:347---357, 2007).

Characterization results on small blocking sets of the polar spaces Q +(2n + 1, 2) and Q +(2n + 1, 3)

In De Beule and Storme, Des Codes Cryptogr 39(3):323–333, De Beule and Storme characterized the smallest blocking sets of the hyperbolic quadrics Q +(2n + 1, 3), n ≥ 4; they proved that these blocking sets are truncated cones over the unique ovoid of Q +(7, 3). We continue this research by classifying all the minimal blocking sets of the hyperbolic quadrics Q +(2n + 1, 3), n ≥ 3, of size at most 3n + 3n–2. This means that the three smallest minimal blocking sets of Q +(2n + 1, 3), n ≥ 3, are now classified. We present similar results for q = 2 by classifying the minimal blocking sets of Q +(2n + 1, 2), n ≥ 3, of size at most 2n + 2n-2. This means that the two smallest minimal blocking sets of Q +(2n + 1, 2), n ≥ 3, are classified.