Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. ...Perfect nonlinear monomials were employed to construct optimal ternary cyclic codes with parameters 3 m -1, 3 m -1-2m, 4 by Carlet, Ding, and Yuan in 2005. In this paper, almost perfect nonlinear monomials, and a number of other monomials over GF(3 m ) are used to construct optimal ternary cyclic codes with the same parameters. Nine open problems on such codes are also presented.
In this paper, a class of permutation trinomials of Niho type over finite fields with even characteristic is further investigated. New permutation trinomials from Niho exponents are obtained from ...linear fractional polynomials over finite fields, and it is shown that the presented results are the generalizations of some earlier works.
The cross-correlation between two maximum length sequences (<inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula>-ary <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula>-sequences takes on the value −1 for at least one shift provided that the decimation <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula> obeys <inline-formula> <tex-math notation="LaTeX">d\equiv 1\,({\mathrm{ mod}}\, p-1) </tex-math></inline-formula>. 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., <inline-formula> <tex-math notation="LaTeX">d=s(p^{n/{2}}-1)+1 </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">s </tex-math></inline-formula> is an integer; (2) all the complete permutation polynomial (CPP) exponents <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula> satisfying <inline-formula> <tex-math notation="LaTeX">d\equiv 1\, ({\mathrm{ mod}}\, p-1) </tex-math></inline-formula>; 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 <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">d=1+l{(p^{rtm}-1)}/{(r+1)} </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">{\mathbb F}_{p^{rtm}} </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula> is a prime, <inline-formula> <tex-math notation="LaTeX">r+1 </tex-math></inline-formula> is an odd prime satisfying <inline-formula> <tex-math notation="LaTeX">p^{r/{2}} \equiv -1\,({\mathrm{ mod}}\, r+1) </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula> is an odd integer (<inline-formula> <tex-math notation="LaTeX">t>2 </tex-math></inline-formula> if <inline-formula> <tex-math notation="LaTeX">p=2 </tex-math></inline-formula>) with <inline-formula> <tex-math notation="LaTeX">\gcd (t,r)=1 </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> is a positive integer. We transform the problem of determining whether <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula> is a CPP exponent into that of investigating the existence of irreducible polynomials over <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{p} </tex-math></inline-formula> with degree <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula> is a CPP exponent over <inline-formula> <tex-math notation="LaTeX">{\mathbb F}_{p^{rtm}} </tex-math></inline-formula> for sufficiently large <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>. When <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> is odd, our new CPP exponents are of Niho type; thus, we give a new class of CPP exponents of Niho type. When <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">d=1+l{(p^{rtm}-1)}/{(r+1)} </tex-math></inline-formula> when <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula> is a sufficiently large integer.
Motivated by recent results on the constructions of permutation polynomials with few terms over the finite field
𝔽
2
n
, where
n
is a positive even integer, we focus on the construction of ...permutation trinomials over
𝔽
2
n
from Niho exponents. As a consequence, several new classes of permutation trinomials over
𝔽
2
n
are constructed from Niho exponents based on some subtle manipulation of solving equations with low degrees over finite fields.
Boomerang connectivity table is a new tool to characterize the vulnerability of cryptographic functions against boomerang attacks. Consequently, a cryptographic function is desired to have boomerang ...uniformity as low as its differential uniformity. Based on generalized butterfly structures recently introduced by Canteaut, Duval and Perrin, this paper presents infinite families of permutations of
F
2
2
n
for a positive odd integer
n
, which have the best known nonlinearity and boomerang uniformity 4. Both open and closed butterfly structures are considered. The open butterflies, according to experimental results, appear not to produce permutations with boomerang uniformity 4. On the other hand, from the closed butterflies we derive a condition on coefficients
α
,
β
∈
F
2
n
such that the functions
V
i
(
x
,
y
)
:
=
(
R
i
(
x
,
y
)
,
R
i
(
y
,
x
)
)
,
where
R
i
(
x
,
y
)
=
(
x
+
α
y
)
2
i
+
1
+
β
y
2
i
+
1
and
gcd
(
i
,
n
)
=
1
, permute
F
2
n
2
and have boomerang uniformity 4. In addition, experimental results for
n
=
3
,
5
indicate that the proposed condition seems to cover all such permutations
V
i
(
x
,
y
)
with boomerang uniformity 4.
The automorphism group of the Zetterberg code
Z
of length 17 (also a quadratic residue code) is a rank three group whose orbits on the coordinate pairs determine two strongly regular graphs ...equivalent to the Paley graph attached to the prime 17. As a consequence, codewords of a given weight of
Z
are the characteristic vectors of the blocks of a PBIBD with two associate classes of cyclic type. More generally, this construction of PBIBDs is extended to quadratic residue codes of length
≡
1
(
mod
8
)
,
to the adjacency codes of triangular and lattice graphs, and to the adjacency codes of various rank three graphs. A remarkable fact is the existence of 2-designs held by the quadratic residue code of length 41 for code weights 9 and 10.
Let p be a prime, n = 2m and d = 3p m - 2 with m ≥ 2, and gcd(d, p n - 1) = 1. In this paper, the correlation distribution between a p-ary m-sequence of period p n - 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 F pn . 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 F 3n , 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.
In this paper, let n = 2k and d = 3 · 2 k - 2 with k ≥ 3 and gcd(d, 2 n - 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 2 n - 1 and its d-decimation sequence is completely determined. This solves a ten-year-old open problem proposed by Dobbertin et al.
Let
p
be a prime number. Reducible cyclic codes of rank 2 over
Z
p
m
are shown to have exactly two Hamming weights in some cases. Their weight distribution is computed explicitly. When these codes ...are projective, the coset graphs of their dual codes are strongly regular. The spectra of these graphs are determined.
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 Z 4 -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.
PDF
