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?Cdifference?? 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).     

Covering arrays from m-sequences and character sums

A covering array of strength t on v symbols is an array with the property that, for every t-set of column vectors, every one of the \(v^t\) possible t-tuples of symbols appears as a row at least once in the sub-array defined by these column vectors. Arrays constructed using m-sequences over a finite field possess many combinatorial properties and have been used to construct various combinatorial objects; see the recent survey Moura et al. (Des Codes Cryptogr 78(1):197–219, 2016). In this paper we construct covering arrays whose elements are the remainder of the division by some integer of the discrete logarithm applied to selected m-sequence elements. Inspired by the work of Colbourn (Des Codes Cryptogr 55(2–3):201–219, 2010), we prove our results by connecting the covering array property to a character sum, and we evaluate this sum by taking advantage of the balanced way in which the m-sequence elements are distributed. Our results include new infinite families of covering arrays of arbitrary strength.     

The multicovering radius problem for some types of discrete structures

The covering radius problem is a question in coding theory concerned with finding the minimum radius r such that, given a code that is a subset of an underlying metric space, balls of radius r over its code words cover the entire metric space. Klapper (IEEE Trans. Inform. Theory 43:1372–1377, 1997) introduced a code parameter, called the multicovering radius, which is a generalization of the covering radius. In this paper, we introduce an analogue of the multicovering radius for permutation codes (Des. Codes Cryptogr. 41:79–86, cf. 2006) and for codes of perfect matchings (cf. 2012). We apply probabilistic tools to give some lower bounds on the multicovering radii of these codes. In the process of obtaining these results, we also correct an error in the proof of the lower bound of the covering radius that appeared in (Des. Codes Cryptogr. 41:79–86, cf. 2006). We conclude with a discussion of the multicovering radius problem in an even more general context, which offers room for further research.     

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.     

Admissible parameters of symmetric designs satisfying v=4(k-\lambda )+2 and symmetric designs with inner balance

The admissible parameters of symmetric \((v,k,\lambda )\) designs satisfying \(v=4(k-\lambda )+2\) are shown to correspond with the solutions of a certain Pell equation. We then determine the feasible parameters of such designs that could have a quasi-symmetric residual design with respect to a block, and classify them into two possible families. Finally, we consider the feasible parameters of symmetric designs with inner balance as defined by Nilson and Heidtmann (Des. Codes Cryptogr. doi:10.1007/s10623-012-9730-2, (2012)), and show that (with one exception) they must all belong to one of these families.     

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 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.     

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]).     

On isotopisms and strong isotopisms of commutative presemifields

In this paper we prove that the P(q,?) (q odd prime power and ?>1 odd) commutative semifields constructed by Bierbrauer (Des. Codes Cryptogr. 61:187?C196, 2011) are isotopic to some commutative presemifields constructed by Budaghyan and Helleseth (SETA, pp.?403?C414, 2008). Also, we show that they are strongly isotopic if and only if q??1(mod?4). Consequently, for each q???1(mod?4) there exist isotopic commutative presemifields of order q ^2? (?>1 odd) defining CCZ-inequivalent planar DO polynomials.     

Improved Agreeing-Gluing Algorithm

Asymptotical complexity of solving a system of sparse algebraic equations over finite fields is studied here. An equation is called sparse if it depends on a bounded number of variables. Finding efficiently solutions to the system of such equations is an underlying hard problem in the cryptanalysis of modern ciphers. New deterministic Improved Agreeing-Gluing Algorithm is introduced. The expected running time of the algorithm on uniformly random instances of the problem is rigorously estimated. The estimate is at present the best theoretical bound on the complexity of solving average instances of the problem. In particular, this is a significant improvement over those in our earlier papers (Semaev, Des Codes Cryptogr 49:47–60, 2008; Semaev, SIAM J Comput 39:388–409 2009). In sparse Boolean equations a gap between the present worst case and the average time complexity of the problem has significantly increased. We formulate Average Time Complexity Conjecture. If proved that will have far-reaching consequences in the field of cryptanalysis and in computing in general.     

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).     

Ovoidal blocking sets and maximal partial ovoids of Hermitian varieties

In Mazzocca et al. (Des. Codes Cryptogr. 44:97–113, 2007), large minimal blocking sets in PG(3, q ²) and PG(4, q ²) have been constructed starting from ovoids of PG(3, q), Q(4, q) and Q(6, q). Some of these can be embedded in a Hermitian variety as maximal partial ovoids. In this paper, the geometric conditions assuring these embeddings are established.     

A recursive construction for simple <Emphasis Type="Italic">t</Emphasis>-designs using resolutions

This work presents a recursive construction for simple t-designs using resolutions of the ingredient designs. The result extends a construction of t-designs in our recent paper van Trung (Des Codes Cryptogr 83:493–502, 2017). Essentially, the method in van Trung (Des Codes Cryptogr 83:493–502, 2017) describes the blocks of a constructed design as a collection of block unions from a number of appropriate pairs of disjoint ingredient designs. Now, if some pairs of these ingredient t-designs have both suitable s-resolutions, then we can define a distance mapping on their resolution classes. Using this mapping enables us to have more possibilities for forming blocks from those pairs. The method makes it possible for constructing many new simple t-designs. We give some application results of the new construction.     

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).      

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.     

An Assmus–Mattson theorem for codes over commutative association schemes

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 \(\mathbb {Z}_4\)-linear codes due to Tanabe (Des Codes Cryptogr 30:169–185, 2003) and Shin et al. (Des Codes Cryptogr 31:75–92, 2004), as well as the original theorem by Assmus and Mattson (J Comb Theory 6:122–151, 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.     

On second-order nonlinearity and maximum algebraic immunity of some bent functions in \mathcal{PS}^{+}

The rth-order nonlinearity and algebraic immunity of Boolean function play a central role against several known attacks on stream and block ciphers. Since its maximum equals the covering radius of the rth-order Reed-Muller code, it also plays an important role in coding theory. The computation of exact value or high lower bound on the rth-order nonlinearity of a Boolean function is very complected/challenging problem, especially when r>1. In this article, we identify a subclass of \({\mathcal{D}}_{0}\) type bent functions constructed by modifying well known Dillon functions having sharper bound on their second-order nonlinearity. We further, identify a subclass of bent functions in \({\mathcal {PS}}^{+}\) class with maximum possible algebraic immunity. The result is proved by using the well known conjecture proposed by Tu and Deng (Des. Codes Cryptogr. 60(1):1–14, 2011). To obtain rth-order nonlinearity (r>2), that is, whole nonlinearity profile of the constructed bent functions is still an open problem.     

On intriguing sets of finite symplectic spaces

Some constructions of intriguing sets of finite symplectic spaces are provided. In particular an affirmative answer to an existence question about small tight sets posed in De Beule et al. (Des Codes Cryptogr 50(2):187–201, 2009) is given.