Cyclic Hadamard Difference Sets Research Articles

A trace representation of binary Jacobi sequences

We determine the trace function representation, or equivalently, the Fourier spectral sequences of binary Jacobi sequences of period p q , where p and q are two distinct odd primes. This includes the twin-prime sequences of period p ( p + 2 ) whenever both p and p + 2 are primes, corresponding to cyclic Hadamard difference sets.

The decimation-Hadamard transform of two-level autocorrelation sequences

A new method to study and search for two-level autocorrelation sequences for both binary and nonbinary cases is developed. This method iteratively applies two operations: decimation and the Hadamard transform based on general orthogonal functions, referred to as the decimation-Hadamard transform (DHT). The second iterative DHT can transform one class of such sequences into another inequivalent class of such sequences, a process called realization. The existence and counting problems of the second iterative DHT are discussed. Using the second iterative DHT, and starting with a single binary m-sequence (when n is odd), we believe one can obtain all the known two-level autocorrelation sequences of period 2/sup n/-1 which have no subfield factorization. We have verified this for odd n/spl les/17. Interestingly, no previously unknown examples were found by this process for any odd n/spl les/17. This is supporting evidence (albeit weak) for the conjecture that all families of cyclic Hadamard difference sets of period 2/sup n/-1 having no subfield factorization are now known, at least for odd n. Experimental results are provided.

Enumeration and criteria for cyclically shift-distinct GMW sequences

Gordon-Mills-Welch (GMW) sequences (also called cascaded GMW sequences) have two-level autocorrelations. This property makes them widely used in various communication and cryptographic systems. The generation of q-ary GMW sequences of period q/sup n-1/ involves three types of parameters. To determine whether GMW sequences are cyclically shift-distinct for differing parameters has remained an open question until now. In this paper, we completely solve this problem for varying all three types of parameters. We find a criterion for cyclically shift-distinct q-ary GMW sequences of period q/sup n-1/, and obtain the number of such sequences. For the special case of q=2, this solution facilitates counting the number of cyclic Hadamard difference sets which correspond to binary GMW sequences of period 2/sup n-1/.

Existence of cyclic Hadamard difference sets and its relation to binary sequences with ideal autocorrelation

Balanced binary sequences with ideal autocorrelation are equivalent to (v, k, λ)-cyclic Hadamard difference sets with v = 4n − 1, k = 2n − 1, λ = n − 1 for some positive integer n. Every known cyclic Hadamard difference set has one of the following three types of v : (1) v = 4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> − 1 is a prime. (2) v is a product of twin primes. (3) v = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> − 1 for n = 2, 3, …. It is conjectured that all cyclic Hadamard difference sets have parameter v which falls into one of the three types. The conjecture has been previously confirmed for n < 10000 except for 17 cases not fully investigated. In this paper, four smallest cases among these 17 cases are examined and the conjecture is confirmed for all v ≤ 3435. In addition, all the inequivalent cyclic Hadamard difference sets with v = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> − 1 for n ≤ 10 are listed and classified according to known construction methods.

Binary pseudorandom sequences of period 2/sup n/-1 with ideal autocorrelation

In this correspondence, we present five new classes of binary sequences of period 2/sup n/-1 with ideal autocorrelation. These sequences, which correspond to new cyclic Hadamard difference sets, were found by extensive computer search. Conjectures on the general construction of these sequences are formulated.

A conjecture on the existence of cyclic Hadamard difference sets

If there exists a cyclic Hadamard difference set of length v, then v = 4 n − 1 is conjectured to be either a prime, or a product of “twin primes”, or one less than a power of 2.

On the existence of cyclic Hadamard difference sets

The main conjecture of this article is the following: if a cyclic (v=4n-1, k=2n-1, /spl lambda/=n-1) Hadamard difference set exists, the the value of v must be either a prime, or a product of twin primes, or one less than a power of 2. Six cases, v=399, 495, 627, 651, 783, and 975, which were once listed as the possible exceptions for v >