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Abstract: Abstract A code C in a generalised quadrangle \({\mathcal {Q}}\) is defined to be a subset of the vertex set of the point-line incidence graph \({\Gamma }\) of \({\mathcal {Q}}\) . The minimum distance \(\delta \) of C is the smallest distance between a pair of distinct elements of C. The graph metric gives rise to the distance partition \(\{C,C_1,\ldots ,C_\rho \}\) , where \(\rho \) is the maximum distance between any vertex of \({\Gamma }\) and its nearest element of C. Since the diameter of \({\Gamma }\) is 4, both \(\rho \) and \(\delta \) are at most 4. If \(\delta =4\) then C is a partial ovoid or partial spread of \({\mathcal {Q}}\) , and if, additionally, \(\rho =2\) then C is an ovoid or a spread. A code C in \({\mathcal {Q}}\) is neighbour-transitive if its automorphism group acts transitively on each of the sets C and \(C_1\) . Our main results (i) classify all neighbour-transitive codes admitting an insoluble group of automorphisms in thick classical generalised quadrangles that correspond to ovoids or spreads, and (ii) give two infinite families and six sporadic examples of neighbour-transitive codes with minimum distance \(\delta =4\) in the classical generalised quadrangle \({\mathsf {W}}_3(q)\) that are not ovoids or spreads. PubDate: 2022-05-21
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Abstract: Abstract 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 and can easily be described. PubDate: 2022-05-21
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Abstract: Abstract We previously proposed the first nontrivial examples of a code having support t-designs for all weights obtained from the Assmus–Mattson theorem and having support \(t'\) -designs for some weights with some \(t'>t\) . This suggests the possibility of generalizing the Assmus–Mattson theorem, which is very important in design and coding theory. In the present paper, we generalize this example as a strengthening of the Assmus–Mattson theorem along this direction. As a corollary, we provide a new characterization of the extended Golay code \({\mathcal {G}}_{24}\) . PubDate: 2022-05-10
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Abstract: Abstract The Korkine–Zolotareff (KZ) reduction and its generalisations, are widely used lattice reduction strategies in communications and cryptography. The KZ constant and Schnorr’s constant were defined by Schnorr in 1987. The KZ constant can be used to quantify some useful properties of KZ reduced matrices. Schnorr’s constant can be used to characterize the output quality of his block 2k-reduction and is used to define his semi block 2k-reduction, which was also developed in 1987. Hermite’s constant, which is a fundamental lattice constant, has many applications, such as bounding the length of the shortest nonzero lattice vector and the orthogonality defect of lattices. Rankin’s constant was introduced by Rankin in 1953 as a generalization of Hermite’s constant. It plays an important role in characterizing the output quality of block-Rankin reduction, proposed by Gama et al. in 2006. In this paper, we first develop a linear upper bound on Hermite’s constant and then use it to develop an upper bound on the KZ constant. These upper bounds are sharper than those obtained recently by the authors, and the ratio of the new linear upper bound to the nonlinear upper bound, developed by Blichfeldt in 1929, on Hermite’s constant is asymptotically 1.0047. Furthermore, we develop lower and upper bounds on Schnorr’s constant. The improvement to the lower bound over the sharpest existing one developed by Gama et al. is around 1.7 times asymptotically, and the improvement to the upper bound over the sharpest existing one which was also developed by Gama et al. is around 4 times asymptotically. Finally, we develop lower and upper bounds on Rankin’s constant. The improvements of the bounds over the sharpest existing ones, also developed by Gama et al., are exponential in the parameter defining the constant. PubDate: 2022-05-09
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Abstract: Abstract An \((r,\ell )\) -good polynomial is a polynomial of degree \(r+1\) that is constant on \(\ell \) subsets of \(\mathbb F_q\) , each of size \(r+1\) . For any positive integer \(r\le 4\) we provide an \((r,\ell )\) -good polynomial such that \(\ell =C_rq+O(\sqrt{q})\) , with \(C_r\) maximal. This directly provides an explicit estimate (up to an error term of \(O(\sqrt{q})\) , with explict constant) for the maximal length and dimension of a Tamo–Barg LRC. Moreover, we explain how to construct good polynomials achieving these bounds. Finally, we provide computational examples to show how close our estimates are to the actual values of \(\ell \) , and we explain how to obtain the best possible good polynomials in degree 5. Our results complete the study by Chen et al. (Des Codes Cryptogr 89(7):1639–1660, 2021), providing \((r,\ell )\) -good polynomials of degree up to 5, with \(\ell \) maximal (up to an error term of \(\sqrt{q}\) ), and our methods are independent. PubDate: 2022-05-07
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Abstract: Abstract Motivated by recent works on the butterfly structure, particularly by its generalization introduced by Canteaut et al. (IEEE Trans Inf Theory 63(11):7575–7591, 2017), we first push further the study of permutation polynomials over binary finite fields by completely characterizing those permutations \(f_{\underline{\epsilon }}\) defined over the finite field \({\mathbb F}_{Q^2}\) (of order \(Q^2\) ) having the following shape: $$\begin{aligned} f_{\underline{\epsilon }}(X):=\epsilon _1\overline{X}^{q+1} +\epsilon _2\overline{X}^qX+\epsilon _3\overline{X}X^q+\epsilon _4X^{q+1} \end{aligned}$$ where \(q=2^k\) , \(Q=2^m\) , m is odd, \(\mathrm {gcd}(m,k)=1\) , \(\overline{X}=X^Q\) and \(\underline{\epsilon }=(\epsilon _1, \epsilon _2, \epsilon _3, \epsilon _4)\in {\mathbb F}_{Q}^4\) . We shall provide an approach to handle the bijectivity of \(f_{\underline{\epsilon }}\) for any \(k\ge 1\) . Notably, we show that the problem of finding conditions for bijectivity of the quadrinomial \(f_{\underline{\epsilon }}\) is closely related to the study of the famous equation \(X^{q+1}+X+a=0\) (*). We then reduce the initial problem into the problem of finding conditions for which an equation of the form (*) has a unique solution in \({\mathbb F}_{Q}\) for every \(a\in {\mathbb F}_{Q}\) . In addition, as a crucial direct consequence our result, we prove the validity of the conjecture (Conjecture 19) proposed by Li et al. (Des Codes Cryptogr 89:737–761, 2021). We emphasize that our positive answer completely characterizes permutations with boomerang uniformity 4 from the butterfly structure, which leads to the view of the quadrinomial \(f_{\underline{\epsilon }}\) as excellent candidates to design block ciphers in symmetric cryptography. Despite a lot of attention regarding the considered conjecture, it remains unsolved on its whole when the coefficients lie in \({\mathbb F}_{Q}\) . However, this article is the first which propose an approach that solves the enter conjecture by handling both sides of it involving equivalence simultaneously. We believe that our novel approach and its strength could benefit from proving the bijectivity of other families of polynomials over finite fields. PubDate: 2022-05-07
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Abstract: Abstract Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal \(q^2\) -ary linear codes. This result can be generalized to \(q^{2 m}\) -ary linear codes, \(m > 1\) . We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to Grassl (Bounds on the minimum distance of linear codes, http://www.codetables.de, 2020) and new q-ary ones, with \(q \ne 2\) , improving others in the literature. PubDate: 2022-05-01
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Abstract: Abstract Zero-divisor codes are codes constructed using group rings where their generators are zero-divisors. Generally, zero-divisor codes can be equivalent despite their associated groups are non-isomorphic, leading to the proposed conjecture “Every dihedral zero-divisor code has an equivalent form of cyclic zero-divisor code”. This paper is devoted to study equivalence of zero-divisor codes in \(F_2G\) having generators from the 2-nilradical of \(F_2G\) , consisting of all nilpotents of nilpotency degree 2 of \(F_2G\) . Essentially, algebraic structures of 2-nilradicals are first studied in general for both commutative and non-commutative \(F_2G\) before specialized into the case when G is cyclic and dihedral. Then, results are used to study the conjecture above in the cases where the codes generators are from their respective 2-nilradicals. PubDate: 2022-05-01
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Abstract: Abstract Recently, b-symbol codes are proposed to protect against b-symbol errors in b-symbol read channels. It is an interesting subject of study to consider the complete b-symbol weight distribution of cyclic codes since b-symbol metric is a generalization for Hamming metric. The complete b-symbol Hamming weight distribution of irreducible cyclic codes is known in only a few cases. In this paper, we give a complete b-symbol Hamming weight distribution of a class of irreducible cyclic codes with two nonzero b-symbol Hamming weights. PubDate: 2022-05-01
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Abstract: Abstract For various quasi-synchronous (QS) CDMA systems, to reduce or eliminate the multiple access interference and multipath interference, it is required to design a set of spreading sequences which are mutually orthogonal within a designed shift zone. In this article, we demonstrate that a concept of irregular spatial assignment, with flexibility to assign different number of users to different cells, can be used to provide the maximal number of orthogonal sequences in any three adjacent cells in networks with a regular tessellation of hexagonal cells. We first consider p-phase spreading sequences of length \(p^m\) (thus nonbinary p-valued sequences) suitable for synchronous (S)-CDMA applications, for \(p>3\) , and give an efficient design method for reaching the maximal cardinality achievable (being \(p^m\) ). A simple solution for a flexible assignment of our orthogonal sets of spreading sequences to the cells in hexagonal networks is given. To address QS-CDMA applications as well, an efficient method to combine these orthogonal sequences with Zadeoff–Chu sequences is proposed for the purpose of designing sets of zero correlation zone (ZCZ) sequences (within a certain shift zone) with optimal parameters, thus reaching the Tang–Fan–Matsufuji bound. A similar design framework, based on the use of some special classes of Boolean functions, is then employed for the binary case to provide the maximum cardinality of pairwise orthogonal sequences of length \(2^m\) through this irregular spatial assignment. This improves upon the best known results achieved in Zhang et al. (IEEE Trans Inf Theory 62:3757–3767, 2016), which assigns \(2^{m-2}\) orthogonal sequences (users) per cell, by doubling the number of users in one third of the network. PubDate: 2022-05-01
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Abstract: Abstract We consider the class of generalized algebraic geometry codes (GAG codes) formed by two collections of places, with places of the same degree in each collection. We introduce the concept of \(N_1N_2\) -automorphism group of a GAG code in this class-that is, a subgroup of the automorphism group of the code. Then we determine a subgroup of the \(N_1N_2\) -automorphism group in the general case and the \(N_1N_2\) -automorphism group itself in the rational function field case. We also explicitly construct such a group. This paper presents a method to obtain similar results for the GAG codes that have more collections of places of the same degree in their construction. PubDate: 2022-04-30
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Abstract: Abstract Arguably, the most important open problem in the theory of q-analogs of designs is the question regarding the existence of a q-analog D of the Fano plane. As of today, it remains undecided for every single prime power order q of the base field. A point P is called an \(\alpha \) -point of D if the derived design of D in P is a geometric spread. In 1996, Simon Thomas has shown that there always exists a non- \(\alpha \) -point. For the binary case \(q = 2\) , Olof Heden and Papa Sissokho have improved this result in 2016 by showing that the non- \(\alpha \) -points must form a blocking set with respect to the hyperplanes. In this article, we show that a hyperplane consisting only of \(\alpha \) -points implies the existence of a partition of the symplectic generalized quadrangle W(q) into spreads. As a consequence, the statement of Heden and Sissokho is generalized to all primes q and all even values of q. PubDate: 2022-04-27
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Abstract: Abstract Minimal codes are characterized by the property that none of the codewords is covered by some other linearly independent codeword. We first show that the use of a bent function g in the so-called direct sum of Boolean functions \(h(x,y)=f(x)+g(y)\) , where f is arbitrary, induces minimal codes. This approach gives an infinite class of minimal codes of length \(2^n\) and dimension \(n+1\) (assuming that \(h: {\mathbb {F}}_2^n \rightarrow {\mathbb {F}}_2\) ), whose weight distribution is exactly specified for certain choices of f. To increase the dimension of these codes with respect to their length, we introduce the concept of non-covering permutations (referring to the property of minimality) used to construct a bent function g in s variables, which allows us to employ a suitable subspace of derivatives of g and generate minimal codes of dimension \(s+s/2+1\) instead. Their exact weight distribution is also determined. In the second part of this article, we first provide an efficient method (with easily satisfied initial conditions) of generating minimal \([2^n,n+1]\) linear codes that cross the so-called Ashikhmin–Barg bound. This method is further extended for the purpose of generating minimal codes of larger dimension \(n+s/2+2\) , through the use of suitable derivatives along with the employment of non-covering permutations. To the best of our knowledge, the latter method is the most general framework for designing binary minimal linear codes that violate the Ashikhmin–Barg bound. More precisely, for a suitable choice of derivatives of \(h(x,y)=f(x) + g(y)\) , where g is a bent function and f satisfies certain minimality requirements, for any fixed f, one can derive a huge class of non-equivalent wide binary linear codes of the same length by varying the permutation \(\phi \) when specifying the bent function \(g(y_1,y_2)=\phi (y_2)\cdot y_1\) in the Maiorana–McFarland class. The weight distribution is given explicitly for any (suitable) f when \(\phi \) is an almost bent permutation. PubDate: 2022-04-11
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Abstract: Abstract Linear codes have widespread applications in data storage systems. There are two major contributions in this paper. We first propose infinite families of optimal or distance-optimal linear codes over \({\mathbb F}_p\) constructed from projective spaces. Moreover, a necessary and sufficient condition for such linear codes to be Griesmer codes is presented. Secondly, as an application in data storage systems, we investigate the locality of the linear codes constructed. Furthermore, we show that these linear codes are alphabet-optimal locally repairable codes with locality 2. PubDate: 2022-04-09
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Abstract: Abstract Two-dimensional (2-D) Z-complementary array pair (ZCAP) is a pair of 2-D arrays, whose 2-D autocorrelation sum gives zero value at all time shifts in a zone around the (0, 0) time shift, except the (0, 0) time shift. The zone is called a zero correlation zone (ZCZ). 2-D ZCAPs include 2-D Golay complementary array pairs (GCAPs) as special cases, and can be applicable in 2-D synchronization. In this paper, we focus on designing new 2-D ZCAPs by exploring two promising approaches. The first construction of 2-D ZCAPs uses 1-D ZCPs as the initial stage, such that any binary ZCP and q-phase ZCP can produce a q-phase 2-D ZCAP. The second construction of 2-D ZCAPs is based on 2-D generalized Boolean functions (GBFs), and the resulting 2-D ZCAPs can have the largest 2-D ZCZ ratio 6/7, compared with known 2-D ZCAPs but not 2-D GCAPs in the literature. Here the ZCZ ratio is defined as the ratio of the ZCZ size over the array size. PubDate: 2022-04-05
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Abstract: Abstract Polycyclic codes are a generalization of cyclic and constacyclic codes. Even though they have been known since 1972 and received some attention more recently, there have not been many studies on polycyclic codes. This paper presents an in-depth investigation of polycyclic codes associated with trinomials. Our results include a number of facts about trinomials, some properties of polycyclic codes, and many new quantum codes derived from polycyclic codes. We also state several conjectures about polynomials and polycyclic codes. Hence, we show useful features of polycyclic codes and present some open problems related to them. PubDate: 2022-04-05
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Abstract: Abstract As the finite Hall planes are Non-Desarguesian, the Pappus Theorem does not hold in them. In this paper we state and prove some weaker versions of Pappus’s Theorem in Hall planes. PubDate: 2022-04-02
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Abstract: Abstract We construct a lattice-based (key-policy) attribute-based signatures (ABS) scheme which supports attributes of unbounded polynomial length (the size of the public parameters is a fixed polynomial in the security parameter and a depth bound, with which one can generate signatures for attributes of arbitrary length). Our scheme does not rely on NIZKs, and we prove that our scheme is semi-adaptively unforgeable in the standard model; that is, the adversary can announce the challenge attribute after seeing the public parameters but before launching any query. Unlike our scheme, previous approaches either construct selectively unforgeable ABS schemes in the standard model that only support attributes of a-priori bounded polynomial length, or construct adaptively unforgeable ABS schemes that support attributes of unbounded polynomial length but relying on NIZKs. We adapt an existing technique developed by Brakerski and Vaikuntanathan for constructing lattice-based semi-adaptively secure (key-policy) attribute-based encryption (ABE) with unbounded attribute length. In particular, we use the adapted technique to generate an unbounded number of matrices out of a-priori bounded public matrices in the construction and program the challenge attribute into the public matrices in our semi-adaptive security proof. Moreover, to achieve adaptive signature query in our semi-adaptive security proof, we employ the traditional partitioning technique developed in identity-based systems to encode the message to be signed. Re-using and adapting lattice-based ABE technique and partitioning technique for lattice-based ABS should not be surprising since the three settings share many features, especially their security proof ideas. PubDate: 2022-04-02
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Abstract: Abstract The substitution box (S-box) is an important nonlinear component in most symmetric cryptosystems and thus should have good properties. Its difference distribution table (DDT) and linear approximation table (LAT) affect the security of the cipher against differential and linear cryptanalysis. In most previous work, differential uniformity and linearity of an S-box are two primary cryptographic properties to impact the resistance against differential and linear attacks. In some cases, the branch number and fixed point are also be considered. However, other important cryptographic properties such as the frequency of differential uniformity (resp. linearity) and the number of Bad Input and Bad Output (BIBO) patterns in DDT (resp. LAT) are often ignored. These properties substantially affect lightweight cryptography based on substitution bit permutation networks (SbPN) such as PRESENT, GIFT and RECTANGLE. This paper introduces a new method to search for S-boxes satisfying all above criteria simultaneously. In our strategy, we transform the process of searching for S-boxes under certain constraints on cryptographic properties into a satisfiability (SAT) problem. As applications, we use our new approach to search out 4-bit and 5-bit S-boxes with the same or better cryptographic properties compared with the S-boxes from well-known ciphers. Finally, we also utilize our method to verify a conjecture proposed by Boura et al. in the case of all 3-bit and 4-bit S-boxes. We propose a proposition and two corollaries to reduce the search space in this verification. PubDate: 2022-04-02
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