Let Z4denote the ring of integers modulo 4. The Galois ring GR(4,m), which consists of 4melements, represents the Galois extension of degreemover Z4. The constructions of codes over Z4have garnered significant interest in recent years. In this paper, building upon previous research, we utilize the defining-set approach to construct several classes of linear codes over Z4by effectively using the properties of the trace function from GR(4,m) to Z4. As a result, we have been able to obtain new infinite families of linear codes over Z4and completely determine their Lee weight distributions.

In this paper, several infinite families of codes over the extension of non-unital non-commutative rings are constructed utilizing general simplicial complexes. Thanks to the special structure of the defining sets, the principal parameters of these codes are characterized. Specially, when the employed simplicial complexes are generated by a single maximal element, we determine their Lee weight distributions completely. Furthermore, by considering the Gray image codes and the corresponding subfield-like codes, numerous of linear codes over Fqare also obtained, whereqis a prime power. Certain conditions are given to ensure the above linear codes are (Hermitian) self-orthogonal in the case ofq= 2; 3; 4. It is noteworthy that most of the derived codes over Fqsatisfy the Ashikhmin-Barg’s condition for minimality. Besides, we obtain two infinite families of distanceoptimal codes over Fqwith respect to the Griesmer bound. By puncturing the Gray image codes and subfield-like codes, several classes of projective codes are presented.

Recently, constructions of optimal linear codes from simplicial complexes have attracted much attention and some related nice works were presented. Let q be a prime power. In this paper, by using the simplicial complexes of${\mathbb {F}}_{q}^{m}$with one single maximal element, we construct four families of linear codes over the ring${\mathbb {F}}_{q}+u{\mathbb {F}}_{q}$($u^{2}=0$), which generalizes the results of Wu et al. (2020). The parameters and Lee weight distributions of these four families of codes are completely determined. Most notably, via the Gray map, we obtain several classes of optimal linear codes over${\mathbb {F}}_{q}$, including (near) Griesmer codes and distance-optimal codes. Moreover, it is shown that most of the Gray images are minimal or self-orthogonal codes which are useful in applications.

Twisted generalized Reed-Solomon (TGRS) codes are an extension of generalized Reed-Solomon (GRS) codes, and have recently attracted significant attention due to their potential for constructing non-GRS MDS codes. This paper presents an in-depth and comprehensive investigation of TGRS codes in their most general form, allowing arbitrary twists at arbitrary positions. First, we introduce a more precise definition of TGRS codes, namely (L,P)-TGRS codes, and provide a concise necessary and sufficient condition for them to be MDS, thereby generalizing previous results. Second, we explicitly characterize the parity check matrices of (L,P)-TGRS codes, and provide a sufficient condition for them to be self-dual. Finally, we investigate the non-GRS properties of (L,P)-TGRS codes via two approaches: the dimensions of Schur squares and combinatorial techniques. As a result, we obtain an infinite family of non-GRS MDS codes.

We propose to study the construction of Boolean bent functions from cyclotomic mappings. By considering Dillon functions, Niho functions and Kasami functions as different branch functions respectively, we obtain three generic constructions from this new perspective. As a result, several infinite classes of bent functions belonging to the${\mathcal {PS}}_{ap}$class, class$\mathcal {H}$and the completed$\mathcal {MM}$class are derived, thereby providing simple representations of known classes of bent functions through cyclotomic mappings. In addition, computer experiments show that examples of bent functions outside these three well-known classes can also be obtained by selecting other branch functions.

In this paper, we construct a large family of projective linear codes over${\mathbb F}_{q}$from the general simplicial complexes of${\mathbb F}_{q}^{m}$via the defining-set construction, which generalizes the results of [IEEE Trans. Inf. Theory 66(11):6762-6773, 2020]. The parameters and weight distributions of this class of codes are completely determined. By using the Griesmer bound, we give a necessary and sufficient condition such that the codes are Griesmer codes and a sufficient condition such that the codes are distance-optimal. For a special case, we also present a necessary and sufficient condition for the codes to be near Griesmer codes. Moreover, by discussing the cases of simplicial complexes with one, two and three maximal elements respectively, the parameters and weight distributions of the codes are given more explicitly, which shows that the codes are at most 2-weight, 5-weight and 19-weight respectively. By studying the optimality of the codes for the three cases in detail, many infinite families of optimal linear codes with few weights over${\mathbb F}_{q}$are obtained, including Griesmer codes, near Griesmer codes and distance-optimal codes.

In this paper, we investigate the power function$F(x)=x^{d}$over the finite field$\mathbb {F}_{2^{4n}}$, where$n$is a positive integer and$d=2^{3n}+2^{2n}+2^{n}-1$. We prove that this power function is AP$c\text{N}$with respect to all$c\in \mathbb {F}_{2^{4n}}\setminus \{1\}$satisfying$c^{2^{2n}+1}=1$, and we determine its$c$-differential spectrum. To the best of our knowledge, this is the second class of AP$c\text{N}$power functions over finite fields of even characteristic. By the same proof ideas, we completely determine the differential spectrum of this function, and give an affirmative answer to a recent conjecture proposed by Budaghyan, Calderini, Carlet, Davidova and Kaleyski.

The cross-correlation between two maximum length sequences ($m$-sequences) of the same period has been studied since the end of 1960s. One open conjecture by Helleseth states that the cross-correlation between any two$p$-ary$m$-sequences takes on the value −1 for at least one shift provided that the decimation$d$obeys$d\equiv 1\,({\mathrm{ mod}}\, p-1)$. This was known as the −1 conjecture. Up to now, the −1 conjecture was confirmed for the following decimations: (1) Niho-type decimations, i.e.,$d=s(p^{n/{2}}-1)+1$, where$s$is an integer; (2) all the complete permutation polynomial (CPP) exponents$d$satisfying$d\equiv 1\, ({\mathrm{ mod}}\, p-1) $; and (3) the additional families of decimations tabulated in this paper. In this paper, we first discuss the connection between the −1 conjecture on cross-correlation of$m$-sequences and CPP exponents, then we confirm the −1 conjecture for a new type of decimations by giving a new class of CPP exponents. The decimations are of the type$d=1+l{(p^{rtm}-1)}/{(r+1)}$over${\mathbb F}_{p^{rtm}}$, where$p$is a prime,$r+1$is an odd prime satisfying$p^{r/{2}} \equiv -1\,({\mathrm{ mod}}\, r+1)$,$t$is an odd integer ($t>2$if$p=2$) with$\gcd (t,r)=1$, and$m$is a positive integer. We transform the problem of determining whether$d$is a CPP exponent into that of investigating the existence of irreducible polynomials over$\mathbb {F}_{p}$with degree$t$satisfying a congruence equation. By a theorem given by Rosen that considered the number of irreducible polynomials with a special congruence relation, we prove that$d$is a CPP exponent over${\mathbb F}_{p^{rtm}}$for sufficiently large$t$. When$m$is odd, our new CPP exponents are of Niho type; thus, we give a new class of CPP exponents of Niho type. When$m$is even, we obtain a new class of CPP exponents which are not of Niho type. As a consequence, we show that the −1 conjecture is true for$d=1+l{(p^{rtm}-1)}/{(r+1)}$when$t$is a sufficiently large integer.

This article focuses on the so-called locally-APN power functions introduced by Blondeau, Canteaut and Charpin, which generalize the well-known notion of APN functions and possibly more suitable candidates against differential attacks. Specifically, given two coprime positive integers$m$and$k$such that$\gcd (2^{m}+1,2^{k}+1)=1$, we investigate the locally-APN-ness property of the Niho type power function$F(x)=x^{s(2^{m}-1)+1}$over the finite field$\mathbb {F}_{2^{2m}}$for$s=(2^{k}+1)^{-1}$, where$(2^{k}+1)^{-1}$denotes the multiplicative inverse modulo$2^{m}+1$. By employing finer studies of the number of solutions of certain equations over finite fields, we prove that$F(x)$is locally-APN and determine its differential spectrum. We emphasize that computer experiments show that this class of locally-APN power functions covers all Niho type locally-APN power functions for$2\leq m\leq 10$. In addition, we also determine the boomerang spectrum of$F(x)$by using its differential spectrum, which particularly generalizes a recent result by Yan, Zhang and Li.

In this paper, we construct four families of linear codes over finite fields from the complements of either the union of subfields or the union of cosets of a subfield, which can produce infinite families of optimal linear codes, including infinite families of (near) Griesmer codes. We also characterize the optimality of these four families of linear codes with an explicit computable criterion using the Griesmer bound and obtain many distance-optimal linear codes. In addition, by a more in-depth discussion on some special cases of these four families of linear codes, we obtain several classes of (distance-)optimal linear codes with few weights and completely determine their weight distributions. It is shown that most of our linear codes are self-orthogonal or minimal which are useful in applications.

Extending previous results, we study a class of general quadrinomials over the field of size 22mwith odd m and characterize conditions under which they are permutations with 4-uniform BCT, a new and important parameter related to boomerang-style attacks. These permutations are known to have the best known nonlinearity. Numerical data also show that the inverse of these functions all have large algebraic degree, making them desirable for applications.

In this paper, a class of p-ary 3-weight linear codes and a class of binary 2-weight linear codes are proposed respectively by virtue of the properties of the perfect nonlinear functions over Fp(m)and (m, s)-bent functions from F2(m)to F2(s), where p is an odd prime and m, s are positive integers. The weight distributions are completely determined by the sign of the Walsh transform of weakly regular bent functions and the size of the preimage of the employed (m, s)-bent functions at the zero point, respectively. As a special case, a class of optimal linear codes meeting Griesmer bound is obtained from our construction.

In Eurocrypt'18, Cid et al. proposed a new cryptanalysis tool called Boomerang Connectivity Table (BCT), to evaluate S-boxes of block ciphers. Later, Boura and Canteaut further investigated the new parameter Boomerang uniformity for cryptographic S-boxes. It is of great interest to find new S-boxes with low Boomerang uniformity for even dimensions. In this paper, we prove that a class of permutation quadrinomials over F2(2m)with m odd has Boomerang uniformity four, which gives the fifth class of such kind of permutation polynomials. Further, the occurrences of 0 and 4 in the BCTs of the investigated permutation polynomials are also completely determined.

An involution over finite fields is a permutation polynomial whose inverse is itself. Owing to this property, involutions over finite fields have been widely used in applications, such as cryptography and coding theory. Following the idea by Wang to characterize the involutory behavior of the generalized cyclotomic mappings, this paper gives a more concise criterion for$x^{r}h(x^{s})\in {\mathbb F} _{q}[x]$being involutions over the finite field${\mathbb F}_{q}$, where$r\geq 1$and$s\,|\, (q-1)$. By using this criterion, we propose a general method to construct involutions of the form$x^{r}h(x^{s})$over${\mathbb F}_{q}$from given involutions over some subgroups of${\mathbb F}_{q}^{*}$by solving congruent and linear equations over finite fields. Then, many classes of explicit involutions of the form$x^{r}h(x^{s})$over${\mathbb F}_{q}$are obtained.

We study the problem of existence of APN functions of algebraic degree n over F2n. We characterize such functions by means of derivatives and power moments of the Walsh transform. We deduce several non-existence results which imply, in particular, that for most of the known APN functions F over F2n. the function x2n-1+ F(x) is not APN, and changing a value of F in a single point then results in non-APN functions. This leads us to conjectures that an APN function modified in one point cannot remain APN and that there exists no APN function of algebraic degree n.

A family IP8of sequences over the 8-ary quadrature-pulse amplitude modulation (Q-PAM) constellation with asymptotically optimal correlation property was presented by Anand and Kumar in 2008. This is the only known family of sequences over the quadrature amplitude modulation (QAM) constellation whose correlation magnitude asymptotically achieves the Welch bound. In this paper, a larger family IP8newof sequences with the same correlation magnitude as that of IP8but double family size is obtained. Moreover, the correlation distributions of IP8and IP8neware also determined for two particular cases in terms of the property of a class of exponential sums over Galois rings. It is also shown that more classes of QAM and Q-PAM sequences with low correlation and larger family size can be obtained from our approach.

Let p be a prime, n = 2m and d = 3pm- 2 with m ≥ 2, and gcd(d, pn- 1) = 1. In this paper, the correlation distribution between a p-ary m-sequence of period pn- 1 and its d-decimation sequence is investigated in a unified approach. Some results for the binary case are extended to the general case. It is shown that the problem of determining the correlation distribution for d can be reduced to that of solving two combinatorial problems related to the unit circle of the finite field Fpn. For an arbitrary odd prime p, it seems difficult to solve these two problems. However, for p = 3, by studying the weight distribution of the ternary Zetterberg code and counting the numbers of solutions of some equations over F3n, the two problems are solved, and thus, the corresponding correlation distribution for d is completely determined. It is noteworthy that this is the first time that the correlation distribution for a non-binary Niho decimation has been determined since 1976.

We study the problem of existence of APN functions of algebraic degree n over F2n. We characterize such functions by means of derivatives and power moments of the Walsh transform. We deduce some non-existence results which mean, in particular, that for most of the known APN functions F over F2nthe function x2n-1+ F(x) is not APN, and changing a value of F in a single point results in non-APN functions.

Linear codes with a few weights have applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper first generalizes the method of constructing two-weight and three-weight linear codes of Ding et al. and Zhou et al. to general weakly regular bent functions and determines the weight distributions of these linear codes. It solves an open problem proposed by Ding et al. Furthermore, this paper constructs new linear codes with two or three weights and presents their weight distributions. They contain some optimal codes meeting certain bound on linear codes.

In this paper, let n = 2k and d = 3 · 2k- 2 with k ≥ 3 and gcd(d, 2n- 1) = 1. Based on some analysis of certain equations over finite fields and the number of codewords with Hamming weight five in Zetterberg code, the correlation distribution between a binary m-sequence of period 2n- 1 and its d-decimation sequence is completely determined. This solves a ten-year-old open problem proposed by Dobbertin et al.

In this paper, we extend two earlier works further in two directions and compute the weight distribution of these cyclic codes under more relaxed conditions. It is interesting to note that many cyclic codes in the family are optimal and have only a few non-zero weights. Besides using similar ideas, we carry out some subtle manipulation of certain exponential sums.

Let m ≥ 3 be an odd integer and p be an odd prime. In this paper, a number of classes of three-weight cyclic codes C(1,e) over Fp, which have parity-check polynomial m1(x)me(x), are presented by examining general conditions on the parameters p, m, and e, where mi(x) is the minimal polynomial of π-i over Fp for a primitive element π of Fpm. Furthermore, for p ≡ 3 (mod 4) and a positive integer e satisfying (pk+ 1) · e ≡ 2 (mod pm - 1) for some positive integer k with gcd(m, k) = 1, the value distributions of the exponential sums T(a, b) = Σx∈FpmωTr(ax+bxe)and S(a, b, c) = Σx∈FpmωTr(ax+bxe+cxs), where s = (pm- 1)/2, are determined. As an application, the value distribution of S(a, b, c) is utilized to derive the weight distribution of the cyclic codes C(1,e,s)with parity-check polynomial m1(x)me(x)ms(x). In the case of p = 3 and even e satisfying the above condition, the dual of the cyclic code C(1,e,s)has optimal minimum distance.

New quadratic bent functions in polynomial form are constructed in this paper. The constructions give new Boolean bent, generalized Boolean bent and p-ary bent functions. Based on Z4-valued quadratic forms, a simple method provides several new constructions of generalized Boolean bent functions. From these generalized Boolean bent functions a method is presented to transform them into Boolean bent and semi-bent functions. Moreover, many new p-ary bent functions can also be obtained by applying similar methods.

In this paper, a construction of codebooks based on a set of bent functions satisfying certain conditions is introduced. It includes some earlier constructions of codebooks meeting the Levenstein bound as special cases. With this construction, two new families of codebooks achieving the Levenstein bound are obtained. The codebooks constructed in this paper could have a very small alphabet size.

In this paper, a class of functions from Niho exponents with four-valued Walsh transform is obtained for any prime by a uniform method, and the distribution of the Walsh transform values is also completely determined. In particular, this class of functions is proven to be bent for a special case. Although it is shown that the obtained bent functions are equivalent to the Leander-Kholosha's class of bent functions, a direct and much simpler proof for the bentness of this kind of Niho functions is provided.

Several new classes of binary andp-ary regular bent functions are obtained in this paper. The bentness of all these functions is determined by some exponential sums over finite fields, most of which have close relations with the well-known Kloosterman sums.

New quadratic bent functions in polynomial forms are constructed in this paper. The constructions give new boolean bent and generalized boolean bent functions. Based on Z4-valued quadratic forms, a simple method provides several new constructions of generalized boolean bent functions. From these generalized boolean bent functions a method is presented to transform them into binary bent and semi-bent functions.

Three classes of binary sequences of period 4Nwith optimal autocorrelation value/magnitude have been constructed by Tang and Gong based on interleaving certain kinds of sequences of periodN, i.e., the Legendre sequence, twin-prime sequence and generalized GMW sequence. In this paper, by means of sequence polynomials of the underlying sequences, the properties of roots of the corresponding sequence polynomials of the interleaved sequences with period 4Nand optimal autocorrelation value/magnitude are discussed in the splitting field ofxN-1 . As a consequence, both the minimal polynomials and linear complexities of these three classes of sequences are completely determined except for the case of the sequences obtained from the generalized GMW sequences. For the latter, the minimal polynomial and linear complexity can be specially obtained if the sequence is constructed based onm-sequences instead of generalized GMW sequences.

Let$m$and$k$be positive integers with$m/{\rm gcd}(m,k)$being odd, for$a\in \BBR$and$b\in \BBL$, the exponential sum$\sum_{x\in \BBL}i^{Tr(ax+2bx^{2^{k}+1})}$is studied systematically in this paper, where$i=\sqrt {-1}$,$\BBR =\BBG \BBR (4,m)$is a Galois ring,$\BBL$is the Teichmüller set of$\BBR$and$Tr(\cdot)$is the trace function from the Galois ring$\BBR$to$\BBZ_{4}$. Through the discussions on the solutions of certain equations and the newly developed theory of$\BBZ_{4}$-valued quadratic forms, the distribution of the exponential sum is completely determined. As its applications, we can determine the Lee weight and Hamming weight distributions of a class of codes${\cal C}^{k}$over$\BBZ_{4}$and the correlation distribution of a quaternary sequence family${\cal U}^{k}$, respectively. Furthermore, the Hamming weight distributions of the binary codes obtained from${\cal C}^{k}$under the most significant bit (MSB) and Gray maps are also determined. For the MSB map sequences of${\cal U}^{k}$, the nontrivial maximal correlation value is given and the correlation distribution is determined for the Gray map sequences of${\cal U}^{k}$. It should be noted that the distribution of the exponential sum for the case$\gcd (m,k)\ne 1$is obtained for the first time, and then the corresponding codes and sequences are novel.

Recently, new optimal Families${\cal S}$and${\cal U}$of quaternary sequences have been presented, and the optimal binary sequence Family${\cal V}$obtained from Family${\cal S}$under Gray map has been investigated as well. The two sequence Families${\cal U}$and${\cal V}$are optimal with respect to the well-known Sidelnikov bound and Welch bound, but their exact correlation distributions are not known until now. In this paper, their exact correlation distributions are completely determined in some cases by making use of exponential sums and the theory of${\bf Z}_4$-valued quadratic forms.

This paper follows the recent work of Helleseth, Kholosha, Johansen, and Ness to study the cross correlation between an m -sequence of period 2m- 1 and the d-decimation of an m-sequence of a shorter period 2n- 1 for an even number m = 2n. Assuming that d satisfies d(2l+ 1) = 2i(mod 2n- 1) for some l > 0 and i > 0, it is proved that the cross correlation takes on either exactly three or four values depending on whether I and n are coprime or not. The distribution of the cross-correlation values is also completely determined. Our results theoretically confirm the numerical data by Ness and Helleseth. It is conjectured that there are no other decimations that give at most four-valued cross correlation apart from the ones proved here.