Abstract: Recently, a class of binary sequences with optimal autocorrelation magnitude has been presented by Su et al. based on Ding-Helleseth-Lam sequences and interleaving technique (Designs, Codes and Cryptography 86, 1329–1338, 2018). The linear complexity of this class of sequences has been proved to be large enough to resist the B-M Algorithm by Fan (Designs, Codes and Cryptography 86, 2441–2450, 2018). In this paper, we study the 2-adic complexities of these sequences with period 4p and show they are no less than 2p, i.e., its 2-adic complexity is large enough to resist the Rational Approximation Algorithm. PubDate: 2019-11-14

Abstract: Determine the number of the rational zeros of any given linearized polynomial is one of the vital problems in finite field theory, with applications in modern symmetric cryptosystems. But, the known general theory for this task is much far from giving the exact number when applied to a specific linearized polynomial. The first contribution of this paper is a better general method to get a more precise upper bound on the number of rational zeros of any given linearized polynomial over arbitrary finite field. We anticipate this method would be applied as a useful tool in many research branches of finite field and cryptography. Really we apply this result to get tighter estimations of the lower bounds on the second-order nonlinearities of general cubic Boolean functions, which has been an active research problem during the past decade. Furthermore, this paper shows that by studying the distribution of radicals of derivatives of a given Boolean function one can get a better lower bound of the second-order nonlinearity, through an example of the monomial Boolean functions \(g_{\mu }=Tr(\mu x^{2^{2r}+2^{r}+1})\) defined over the finite field \({\mathbb F}_{2^{n}}\). PubDate: 2019-11-13

Abstract: Complementary sequences with quadrature amplitude modulation (QAM) symbols have important applications in OFDM communication systems. The objective of this paper is to present two constructions of 16-QAM complementary sequence sets of size 4. The first construction generates four complementary sequences of length L = 2m− 1 + 2v, where m and v are two positive integers with 1 ≤ v ≤ m − 1. The second one leads to four complementary sequences of length L = 2m− 1 + 1. It turns out that the peak-to-mean envelope power ratios (PMEPRs) of constructed complementary sequence sets are upper bounded by 4. PubDate: 2019-11-12

Abstract: Due to the wide applications in communications, data storage and cryptography, linear codes have received much attention in the past decades. As a subclass of linear codes, minimal linear codes can be used to construct secret sharing with nice access structure. The objective of this paper is to construct new classes of minimal binary linear codes with \(w_{\min \limits }/w_{\max \limits }\leq 1/2\) from preferred binary linear codes, where \(w_{\min \limits }\) and \(w_{\max \limits }\) denote the minimum and maximum nonzero Hamming weights in \(\mathcal {C}\) respectively. Firstly, we introduce a concept called preferred binary linear codes and a class of minimal binary linear codes with \(w_{\min \limits }/w_{\max \limits }\leq 1/2\) can be deduced from preferred binary linear codes. As an application of preferred binary linear codes, we get a new class of six-weight minimal binary linear codes with \(w_{\min \limits }/w_{\max \limits }< 1/2\) from a known class of five-weight preferred binary linear codes. Secondly, by employing vectorial Boolean functions, we construct two new classes of preferred binary linear codes and, consequently, these two new classes of preferred binary linear codes can generate two new classes of minimal binary linear codes with \(w_{\min \limits }/w_{\max \limits }\leq 1/2\) and large minimum distance. PubDate: 2019-11-08

Abstract: A quaternary sequence is said to be optimal if its odd-periodic autocorrelation magnitude equal to 2 for even length, and 1 for odd length. In this paper, we propose three constructions of optimal quaternary sequences: the first construction applies the inverse Gray mapping to four component binary sequences, which could be chosen from GMW sequence pair, twin-prime sequence pair, Legendre sequence pair, and ideal sequences; the second one generates optimal sequences from quaternary sequences with optimal even-periodic autocorrelation magnitude; the third one gives new optimal quaternary sequences by applying the sign alternation transform and Gray mapping to GMW sequence pair and twin-prime sequence pair. In particular, some proposed sequences have new parameters. PubDate: 2019-11-08

Abstract: Constacyclic codes are a subclass of linear codes and have been well studied. Constacyclic BCH codes are a family of constacyclic codes and contain BCH codes as a subclass. Compared with the in-depth study of BCH codes, there are relatively little study on constacyclic BCH codes. The objective of this paper is to determine the dimension and minimum distance of a class of q-ary constacyclic BCH codes of length \(\frac {q^{m}-1}{q-1}\) with designed distances \(\delta _{i}=q^{m-1}-\frac {q^{\lfloor \frac {m-3}2 \rfloor +i }-1}{q-1}\) for \(1\leq i\leq \min \limits \{\lceil \frac {m+1}2 \rceil -\lfloor \frac {m}{q+1} \rfloor , \lceil \frac {m-1}2 \rceil \}\) . As will be seen, some of these codes are optimal. PubDate: 2019-11-06

Abstract: Some classes of binary codes constructed by using some defining sets are studied, and for most defining sets, we will determine the generalized Hamming weight of the corresponding codes completely, and for other defining sets, we will determine part of the generalized Hamming weight of the corresponding codes. PubDate: 2019-10-30

Abstract: This paper generalizes three constructions of families of sequences with bounded off peak correlation with application to Code Division Multiple Access (CDMA), frequency hopping, and Ultra Wide Band (UWB). These new families present flexible family sizes and sequence lengths, making them well suited to wireless communications and Multiple Input Multiple Output (MIMO) radar. In particular, we show that the generalized Kamaletdinov I construction offers the lowest off-peak correlation for a given family size. For the two smallest family sizes, this construction asymptotically matches the performance of the small Kasami set and Gold codes, respectively. Among other properties, it has one of the slowest off-peak correlation growth of all known constructions, increasing proportionally to the logarithm of the family size and generates frequency and time hopping sequences. PubDate: 2019-10-08

Abstract: Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study the largest minimum weights d(n,k) among all binary linear complementary dual [n,k] codes. We determine d(n,4) for n ≡ 2,3,4,5,6,9,10,13 (mod 15), and d(n,5) for n ≡ 3,4,5,7,11,19,20, 22,26 (mod 31). Combined with known results, d(n,k) are also determined for n ≤ 24. PubDate: 2019-10-08

Abstract: Let \(R=\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \) . Then R is a local non-principal ideal ring of 16 elements. First, we give the structure of every cyclic code of odd length n over R and obtain a complete classification for these codes. Then we determine the cardinality, the type and its dual code for each of these cyclic codes. Moreover, we determine all self-dual cyclic codes of odd length n over R and provide a clear formula to count the number of these self-dual cyclic codes. Finally, we list some optimal 2-quasi-cyclic self-dual linear codes of length 30 over \(\mathbb {Z}_{4}\) and obtain 4-quasi-cyclic and formally self-dual binary linear [60,30,12] codes derived from cyclic codes of length 15 over \(\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \) . PubDate: 2019-10-07

Abstract: In this paper, we study two special subsets of a finite field of odd characteristics associated with non-weakly regular bent functions. We show that those subsets associated to non-weakly regular even bent functions in the GMMF class (see Çesmelioğlu et al. Finite Fields Appl. 24, 105–117 2013) are never partial difference sets (PDSs), and are PDSs if and only if they are trivial subsets. Moreover, we analyze the two known sporadic examples of non-weakly regular ternary bent functions given in Helleseth and Kholosha (IEEE Trans. Inf. Theory 52(5), 2018–2032 2006, Cryptogr. Commun. 3(4), 281–291 2011). We observe that corresponding subsets are non-trivial partial difference sets. We show that they are the union of some cyclotomic cosets and so correspond to 2-class fusion schemes of a cyclotomic scheme. We also present a further construction giving non-trivial PDSs from certain p-ary functions which are not bent functions. PubDate: 2019-09-14

Abstract: Based on the parity of the number of occurrences of a pattern 10 as a scattered subsequence in the binary representation of integers, a Rudin-Shapiro-like sequence is defined by Lafrance, Rampersad and Yee. The Nth maximum order complexity and the expansion complexity of this Rudin-Shapiro-like sequence are calculated in this paper. PubDate: 2019-09-13

Abstract: In quasi-synchronous frequency-hopping multiple-access systems where relative delays are restricted within a certain zone, low hit zone frequency-hopping sequences (LHZ FHSs) with favorable partial Hamming correlation properties are desirable. In this paper, we present a new class of LHZ FHS sets with optimal partial Hamming correlation based on t-decimation of m-sequence. PubDate: 2019-09-13

Abstract: This paper provides a complete answer to three conjectures of Parker about low odd-periodic autocorrelation of sixteen cyclotomic binary sequences in Parker (2001), and also gives new binary sequences with low odd-periodic autocorrelation. PubDate: 2019-09-10

Abstract: Each finite binary sequence (sh) is associated with a Boolean function B. The correlation measure of order k and the r-th order nonlinearity are figures of merit for the unpredictability of (sh) and B, respectively. We estimate the r-th order nonlinearity of B in terms of the correlation measure of order 2r of (sh). We apply our result to Boolean functions associated with the Legendre sequence, that is, the binary sequence describing the least significant bit of the discrete logarithms in the finite field \(\mathbb {F}_{p}\) of p elements, where p > 2 is a prime. PubDate: 2019-09-01

Abstract: A concatenated construction for linear complementary dual codes was given by Carlet et al. using the so-called isometry inner codes. Here, we obtain a concatenated construction to the more general family, linear complementary pair of codes. Moreover, we extend the dual code description of Chen et al. for concatenated codes to duals of generalized concatenated codes. This allows us to use generalized concatenated codes for the construction of linear complementary pair of codes. PubDate: 2019-09-01

Abstract: In this article a family of statistical randomness tests for binary strings are introduced, based on Golomb’s pseudorandomness postulate R-2 on the number of runs. The basic idea is to construct recursive formulae with computationally tenable probability distribution functions. The technique is illustrated on testing strings of \(2^{7}\) , \(2^{8}\) , \(2^{10}\) and \(2^{12}\) bits. Furthermore, the expected value of the number of runs with a specific length is obtained. Finally the tests are applied to several collections of strings arising from different pseudorandom number generators. PubDate: 2019-09-01

Abstract: In this paper we construct two classes of binary quantum error-correcting codes on closed orientable surfaces. These codes are derived from self-dual orientable embeddings of complete bipartite graphs and complete multipartite graphs on the corresponding closed orientable surfaces. We also show a table comparing the rate of these quantum codes when fixing the minimum distance to 3 and 4. PubDate: 2019-09-01

Abstract: Nowadays it is vital to have a robust mechanism that can identify people, objects, animals, and living beings, for example, for agricultural, health and national security purposes. Some drawbacks occur when very many objects need to be identified, and the tool is unable to support all of them. Even if the mechanism could tag them all, it is also important that the labels or codewords not resemble each other, to be able to detect and correct errors. To solve this problem, this article proposes an MDS (Maximum Distance Separable) code C with length 11 and dimension 7 over the finite field \({\mathbb {F}}_{2^{10}}\) . Furthermore, we construct a subcode of C with capacity for 327 different identifiers. Concretely we consider the set of all codewords with entries belonging to the subfield of \({\mathbb {F}}_{2^{10}}\) isomorphic to \({\mathbb {F}}_{2^{5}}\) . A decoding algorithm and an encryption method using elliptic curves cryptography for the codewords are also proposed. PubDate: 2019-09-01