Let A = (aij)k×n be a matrix with entries in the Galois field GF (2), and let x = (x1, x2, . . . , xn) be a vector of variables assuming values in GF (2). The gate complexity of A, denoted by C(A), is the minimum number of XOR gates necessary to compute the matrix-vector product Ax. In this paper it is shown that C(Hk) = 2k+1 − 2k − 2, where Hk is the parity-check matrix of the [2k − 1, 2k − k ...

will contain infinitely many prime terms — known as Mersenne primes. The Mersenne prime conjecture is related to a classical problem in number theory concerning perfect numbers. A whole number is said to be perfect if, like 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14, it is equal to the sum of all its divisors apart from itself. Euclid pointed out that 2k−1(2k − 1) is perfect whenever 2 − 1 is pr...

A (1,-1)-matrix will be called a bent type matrix if each row and each column are bent sequences. A similar description can be found in Carlisle M. in which the authors use the properties of bent type matrices to construct a class of bent functions. In this paper we give a general method to construct bent type matrices and show that the bent sequence obtained from a bent type matrix is a genera...

MHC class I molecules present a variable but limited repertoire of antigenic peptides for T-cell recognition. Understanding how peptide selection is achieved requires mechanistic insights into the interactions between the MHC I and candidate peptides. We find that, at first encounter, MHC I H-2K(b) considers a wide range of peptides, including those with expanded N termini and unfitting anchor ...

Let Z be the set of integers, and for a prime p let Zp denote the set of those rational numbers whose denominator is not divisible by p. Let (p ) be the Legendre symbol. Suppose that p is an odd prime and a ∈ Zp. In [7] the author investigated congruences for ∑[p/4] k=0 (4k 2k ) ak modulo p, where [x] is the greatest integer not exceeding x. For k ∈ {0, 1, . . . , p − 1} it is easily seen that ...

The secondary cytotoxic responses to the male specific antigen (H-Y) in in mice show H-2 restrictions so that cytotoxic female cells must share K and/or D end antigen with the male target cells. The production of cytotoxic cells is under the control of Ir genes, thus offering the possibility of studying the function of Ir genes in H-2-restricted cytotoxic responses. There are two kinds of Ir ge...

Abstract—The free distance of (n, k, l) convolutional codes has some connection with the memory length, which depends on not only l but also on k. To efficiently obtain a large memory length, we have constructed a new class of (2k, k, l) convolutional codes by (2k, k) block codes and (2, 1, l) convolutional codes, and its encoder and generation function are also given in this paper. With the he...

([3]) and tvs(G) ≤ ⌈ 3n δ ⌉ + 1 ([1]). The exact values for some families of graphs are also known, e.g. the value of s(Cin(1, k), given in [2]. We prove that tvs(Cin(1, 2, . . . , k)) = n+2k 2k+1 , while s(Cin(1, 2, . . . , k)) = n+2k−1 2k . In order to do that, we split the graph Cin(1, 2, . . . , k) into segments and label each segment using 0, 1 and 2 in such a way that the weighted degrees...

Let f(N) and xi(-1)(N) represent, respectively, the free energy per spin and the inverse spin-spin correlation length of the critical Ising model on a N x infinity lattice, with f(N)-->f(infinity) as N-->infinity. We obtain analytic expressions for a(k) and b(k) in the expansions N( f(N)-f(infinity)) = SUM (k = 1)(infinity)a(k)/N(2k-1) and xi(-1)(N) = SUM (k = 1)(infinity)b(k)/N(2k-1) for squar...

A 2-coloring of the n-cube in the n-dimensional Euclidean space can be considered as an assignment of weights of 1 or 0 to the vertices. Such a colored ncube is said to be balanced if its center of mass coincides with its geometric center. Let Bn,2k be the number of balanced 2-colorings of the n-cube with 2k vertices having weight 1. Palmer, Read, and Robinson conjectured that for n ≥ 1, the se...