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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.

Let $n\geq 3$ be an odd integer, $d_{1} = {\frac {3^{n}-1}{2}} -1 $ , $d_{2} =3^{n}-2$ and u be an element of the finite field $\mathbb {F}_{3^{n}}$ . This paper shows that $f_{u}(x)=ux^{d_{1}}+x^{d_{2}}$ is an almost perfect nonlinear (APN) function on $\mathbb {F}_{3^{n}}$ if and only if $\chi (u+1)=\chi (u-1)=\chi (u)$ , where $\chi (\cdot)$ denotes the quadratic character of $\mathbb {F}_{3^{n}}$ . 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 $f_{u}(x)$ for elements u satisfying $\chi (u+1)=\chi (u-1)$ and express them in terms of several quadratic character sums of cubic polynomials.

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.

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.