How to Build Robust Shared Control Systems

Previous researchers have designed shared control schemes with a view to minimising the likelihood that participants will conspire to perform an unauthorised act. But, human nature being what it is, systems inevitably fail; so shared control schemes should also be designed so that the police can identify conspirators after the fact. This requirement leads us to search for schemes with sparse access structures. We show how this can be done using ideas from coding theory. In particular, secret sharing schemes based on geometric codes whose dual [n,k,d] codes have d and n as their only nonzero weights are suitable. We determine their access structures and analyse their properties. We have found almost all of them, and established some relations among codes, designs and secret-sharing schemes.     

Some results about the cross-correlation function between two maximal linear sequences

Let {a₁} and {a_d1} be two maximal linear sequences of period pⁿ ? 1. The cross-correlation function is defined by C_d(t) =

for t = 0, t…pⁿ ? 2, where $\text{ζ = exp(2π}\text{1}\text{p}\text{)}$. We find some new general results about C_d(t). We also determine the values and the number of occurences of each value of C_d(t) for several new values of d.     

New M-ary sequences with low autocorrelation from interleaved technique

Let \(p\) and \(q\) be two odd primes with \(p=Mf+1\) and \(M\) is even. A new construction of \(M\) -ary sequences of period \(pq\) with low periodic autocorrelation is presented in this paper based on interleaving the \(M\) -ary power residue sequence of period \(p\) according to the quadratic residue with respect to \(q\) . This construction can generate the well-known twin-prime sequence and generalized cyclotomy sequence of order two if \(M=2\) . For \(M=4\) , a new class of quaternary sequences of period \(pq\) with maximal nontrivial autocorrelation value being either \(\sqrt{5}\) or \(3\) is obtained. This achieves the best known results for such kind of quaternary sequences.     

Split Weight Enumerators for the Preparata Codes with Applications to Designs

For quaternary Preparata and Kerdock codes of length N=2 ^m ,m odd, we prove that the split complete weight enumerator for a coordinate partition into 3 and N-3 coordinates is independent of the chosen partition. The result implies that the words of a given complete weight in either a Preparata code or Kerdock code define a 3-design.     

On the p-Ranks and Characteristic Polynomials of Cyclic Difference Sets

In this paper, the p-ranks and characteristic polynomials of cyclic difference sets are derived by expanding the trace expressions of their characteristic sequences. Using this method, it is shown that the 3-ranks and characteristic polynomials of the Helleseth–Kumar–Martinsen (HKM) difference set and the Lin difference set can be easily obtained. Also, the p-rank of a Singer difference set is reviewed and the characteristic polynomial is calculated using our approach.     

Construction of Some Optimal Ternary Linear Codes and the Uniqueness of [294, 6, 195; 3]-Codes Meeting the Griesmer Bound

Let n_q(k, d) denote the smallest value of n for which there exists an [n, k, d; q]-code. It is known (cf. (J. Combin. Inform. Syst. Sci.18, 1993, 161–191)) that (1) n₃(6, 195) {294, 295}, n₃(6, 194) {293, 294}, n₃(6, 193) {292, 293}, n₃(6, 192) {290, 291}, n₃(6, 191) {289, 290}, n₃(6, 165) {250, 251} and (2) there is a one-to-one correspondence between the set of all nonequivalent [294, 6, 195; 3]-codes meeting the Griesmer bound and the set of all {v₂ + 2v₃ + v₄, v₁ + 2v₂ + v₃; 5, 3}-minihypers, where v_i = (3ⁱ − 1)/(3 − 1) for any integer i ≥ 0. The purpose of this paper is to show that (1) n₃(6, 195) = 294, n₃(6, 194) = 293, n₃(6, 193) = 292, n₃(6, 192) = 290, n₃(6, 191) = 289, n₃(6, 165) = 250 and (2) a [294, 6, 195; 3]-code is unique up to equivalence using a characterization of the corresponding {v₂ + 2v₃ + v₄, v₁ + 2v₂ + v₃; 5, 3}-minihypers.     

A class of permutation quadrinomials

In this paper, we investigate the permutation behavior of a class of quadrinomials. Each term of these quadrinomials has a Niho-type exponent, and two sets of coefficient triples making the quadrinomials to be permutations are obtained. We use a substitution to transform the permutation problem into the root distribution problem in the unit circle of certain quadratic and cubic equations.     

An Infinite Family of 3-Designs from Preparata Codes overZ4

We show that the support of minimum Lee weight codewords having Hamming weight 5 in the Preparata code over Z₄ form a 3-(2^m,5,10) design for any odd integer m 3.     

Two New Infinite Families of 3-Designs from Kerdock Codes over Z4

In this paper we show that the support of the codewords of each type in the Kerdock code of length 2^m over Z₄ form 3-designs for any odd integer . In particular, twonew infinite families of 3-designs are obtained in this constructionfor any odd integer . In particular, twonew infinite families of 3-designs are obtained in this constructionfor any odd integer , whose parameters are ,and .