For integers u ≥ 2 and s ≥ 1, let q0 = 2u, and let Cs(q0) be the generalized Zetterberg code of length n = qs 0 + 1 over the finite field Fq0 of characteristic 2. For odd characteristic, the covering radius of Cs(q0) was determined recently, whereas the case of even characteristic remained open. In this paper, we determine the covering radius of generalized Zetterberg codes over finite fields of characteristic 2, thereby solving this open problem. Our approach uses methods from the theory of algebraic curves over finite fields. As an application, we obtain an infinite family of quasi-perfect codes.

The existence of optimal binary self-orthogonal codes has been well characterized. In this paper, we develop general methods involving residual codes and the MacWilliams identities to prove the nonexistence of several infinite families of binary self-orthogonal codes, despite the existence of binary linear codes with the same parameters. In particular, we focus on the largest minimum distances of optimal binary self-orthogonal codes with dimension eight.

Two classes of ternary bent functions of degree four with two and three terms in the univariate representation that belong to the completed Maiorana-McFarland class are found. Binomials are mappings _3^4k↦ given by f(x)=_4k(a_1 x^2(3^k+1)+a_2 x^(3^k+1)^2), where a_1 is a nonsquare in _3^4k and a_2 is defined explicitly by a_1. Particular subclasses of the binomial bent functions we found can be represented by exceptional polynomials over . Bent trinomials are mappings _3^2k↦ given by f(x)=_n(a_1 x^2·3^k+4 + a_2 x^3^k+5 + a_3 x^2) with coefficients explicitly defined by the parity of k. The proof is based on a new criterion that allows checking bentness by analyzing first- and second-order derivatives of f in the direction of a chosen n/2-dimensional subspace.

Hyperbent functions are special Boolean functions which composited with any bijective power function have maximal distance from affine functions. This paper is devoted to a class of hyperbent functions having the form f_a,b^d(x)= Tr _1^2n(a(x^2^n-1+b)^d) for a, b∈𝔽_2^2n^* . For a∈𝔽_2^n^* and several exponents d, we characterize parameters a and b, which yield hyperbent functions f_a,b^d(x) , via partial exponential sums of Dickson polynomials; in a particular case when b=1 we uncover that hyperbent functions can be derived from bijective power functions, including almost nonlinear power functions for odd n. When a∈𝔽_2^2n∖𝔽_2^n and b=1 , we introduce a parameterized method to reduce the computation of involved exponential sums which also lead to explicit hyperbent functions.

We propose a new coded space shift keying (CSSK) signaling technique for multi-user (MU), multiple-input multiple-output (MIMO) communication systems incorporating physical layer security (PLS). Besides its error correction ability, the designed linear code is capable of choosing transmit antenna indices automatically and selecting the best set of antenna combinations that minimizes the bit error rate (BER). Results obtained for the single-user (SU) schemes are then extended to a general single-cell downlink MU CSSK setting. A precoder design is proposed with a maximum ratio combining (MRC) technique to eliminate the multi-user interference (MUI) entirely by taking advantage of channel state information (CSI) at the transmitter. It is shown that the same precoding provides a very effective jamming signal for the PLS against passive eavesdroppers, degrading their signal-to-interference-plus-noise ratio (SINR) severely. A closed-form expression for the achievable secrecy rates is derived and it is maximized by the proposed power allocation algorithm. Finally, it is shown analytically and by computer simulations that substantially better BER performance is achieved by each user over interference-free transmission compared to an SU transmission with a maximum likelihood (ML) detector.

Sequences play an important role in communication and radar systems, where related theoretical bounds serve as benchmarks to access the designed sequences. This paper first studies the odd-periodic ambiguity function (OPAF) of sequence sets and derives a theoretical lower bound of its magnitude. Based on the even-odd transformation, the relationship between the periodic ambiguity function and the odd-periodic ambiguity function is established. Furthermore, we construct two classes of odd-periodic sequences that achieve the optimal zero ambiguity zone (ZAZ) by leveraging the properties of known sequences and quadratic functions. Finally, we present a class of low ambiguity zone (LAZ) sequence sets by using certain cubic functions, which is asymptotically optimal with respect to the derived bound.

The symbol-pair coding theory was put forward by Cassuto and Blaum [IEEE TIT, 2011] to be applicable in high-density storage situations. Yaakobi et al. [IEEE TIT, 2016] extended the concept of symbol-pair metric to b-symbol metric when b >= 2. The extensive research on the b-symbol Hamming weight spectra of cyclic codes has been centered on the case where the alphabet is a finite field. The case of cyclic codes over Z(4), an extremely important class of codes, has been overlooked for a long time in the exploration of the b-symbol Hamming weight spectra. In this paper, we study the b-symbol Hamming weight spectra of the shortened Kerdock codes K-m(-) and the Kerdock codes K-m over Z(4). The formulas for calculating the symbol-pair Hamming weight of the codewords in K-m(-) and K-m are given, and their values hinge on the trace-like function values of specific elements in the Teichmuller set. In particular, we present a class of Z(4)-cyclic codes with three non-zero b-symbol Hamming weights. As by-products, the b-symbol Hamming weight hierarchies of the Preparata codes and the Goethals codes are provided.

The purpose of this paper is two-fold. First, we characterize the existence of binary self-orthogonal codes meeting the Griesmer bound by employing Solomon-Stiffler codes and some related residual codes. Second, using such a characterization, we determine the exact value of $d_{so}(n,7)$ except for five special cases and the exact value of $d_{so}(n,8)$ except for 41 special cases, where $d_{so}(n,k)$ denotes the largest minimum distance among all binary self-orthogonal $[n, k]$ codes. Currently, the exact value of $d_{so}(n,k)$ $(k \le 6)$ was determined by Shi et al. (2022). In addition, we develop a general method to prove the nonexistence of some binary self-orthogonal codes by considering the residual code of a binary self-orthogonal code.

For an integer s >= 1, let C-s(q(0)) be the generalized Zetterberg code of length q(0)(s) + 1 over the finite field F-q0 of odd characteristic. Recently, Shi et al. determined the covering radius of C-s(q(0)) for q(0)(s)not equivalent to 7(mod 8), and left the remaining case as an open problem. In this paper, we develop a general technique involving arithmetic of finite fields and algebraic curves over finite fields to determine the covering radius of all generalized Zetterberg codes for q(0)(s)equivalent to 7(mod 8), which therefore solves this open problem. We also introduce the concept of twisted half generalized Zetterberg codes of length q(0)(s)+1/2, and show the same results hold for them. As a result, we obtain some quasi-perfect codes.

Let n be a positive integer, p an odd prime, d = p(n+1)/4 if p(n) equivalent to 3 (mod 8) and d = p(n+1)/4 + p(n-1)/2 if p(n) equivalent to 7 (mod 8). It was shown that the differential uniformity of the power permutation x(d) over F-pn is no more than 4. In this paper, we further investigate the differential properties of this power permutation. By studying the differential equation of x(d) and certain equation system over F-pn, we completely determined the differential spectrum of this power permutation.

Let n be a positive integer, p be an odd prime, d=p^n+1/4+p^n-1/2 if p^n ≡ 3 (mod 8) and d=p^n+1/4 if p^n ≡ 7 (mod 8) . When p^n>7 , the power mapping x^d from 𝔽_p^n to 𝔽_p^n was proved to be almost perfect nonlinear by Helleseth, Rong and Sandberg in IEEE Trans. Inform. Theory, 45(2): 475-485, 1999. By regarding the components in the differential spectrum as unknowns and establishing a system of linear equations concerning them, Tan and Yan completely determined the differential spectrum of this power mapping in Des. Codes Cryptogr., 91(8): 2755-2768, 2023. In this paper, we directly characterize the conditions on b∈𝔽_p^n under which the derivative equation (x+1)^d-x^d=b has exactly i solution(s) for i=0,1,2 , respectively. Then, utilizing the theory of character sums, the number of those b’s in each case is determined and thus the differential spectrum of x^d is obtained. Our method provides more information about the derivative equation of x^d , which can be used to describe the DDT of this APN power function.

Determining the weight distribution of a code is an old and fundamental topic in coding theory that has been thoroughly studied. In 1977, Helleseth, Kløve, and Mykkeltveit presented a weight enumerator polynomial of the lifted code over ${\mathbb {F}}_{q^{\ell } }$ of a q-ary linear code with significant combinatorial properties, which can determine the support weight distribution of this linear code. The Solomon-Stiffler codes are a family of famous Griesmer codes, which were proposed by Solomon and Stiffler in 1965. In this paper, we determine the weight enumerator polynomials of the lifted codes of the projective Solomon-Stiffler codes using some combinatorial properties of subspaces. As a result, we determine the support weight distributions of the projective Solomon-Stiffler codes. In particular, we determine the weight hierarchies of the projective Solomon-Stiffler codes.

Maximum-length sequences (m-sequences for short) over finite fields are generated by linear feedback shift registers with primitive characteristic polynomials. These sequences have nice mathematical structures and good randomness properties that are favorable in practical applications. During the past five decades, the crosscorrelation between m-sequences of the same period has been intensively studied, and a particular research focus has been on investigating the cross-correlation spectra with few possibles values. In this chapter we summarize all known results on this topic in the literature and promote several open problems for future research.

Nonlinear complexity is an important measure for assessing the randomness of sequences. In this paper we investigate how circular shifts affect the nonlinear complexities of finite-length binary sequences and then reveal a more explicit relation between nonlinear complexities of finite-length binary sequences and their corresponding periodic sequences. Based on the relation, we propose two algorithms that can generate all periodic binary sequences with any prescribed nonlinear complexity.

Let n ≥ 3 be an odd integer, d 1 =3 n -1/2-1, d 2 =3 n -2 and u be an element of the finite field F 3n . This paper shows that fu(x)=uxd1+xd2 is an almost perfect nonlinear (APN) function on F 3 n if and only if χ( u +1)= χ( u -1)=χ( u ), where χ(∙) denotes the quadratic character of F3n. This settles the open problem raised by Ness and Helleseth in IEEE Trans. Inf. Theory 53(7): 2581-2586, 2007, where only the sufficiency part of the result was proved. Furthermore, we investigate the differential spectra of fu(x) for elements u satisfying χ( u +1)=χ( u -1) and express them in terms of several quadratic character sums of cubic polynomials.