For any prime p, λ-constacyclic codes of length p over R = Fpm + uFpm are precisely the ideals of the local ring Rλ = R[x] 〈xp−λ〉 , where u = 0. In this paper, we first investigate the Hamming distances of cyclic codes of length p over R. The minimum Hamming distances of all cyclic codes of length p over R are determined. Moreover, an isometry between cyclic and α-constacyclic codes of length p...

This thesis investigates uniform mixing on Cayley graphs over Z3. We apply Mullin’s results on Hamming quotients, and characterize the 2(d+2)-regular connected Cayley graphs over Z3 that admit uniform mixing at time 2π/9. We generalize Chan’s construction on the Hamming scheme H(d, 2) to the schemeH(d, 3), and find some distance graphs of the Hamming graph H(d, 3) that admit uniform mixing at t...

This work presents a brief survey on codes (with respect to Hamming metric) which are perfect-like i.e. codes which are capable to correct a given set of error vectors and no others. Several kinds of such codes which exist in the literature have been discussed and possibilities of the existence of such codes in some cases have been explored. 2000 Mathematics Subject Classification: 94B25, 94B60...

Hoffstein and Silverman suggested a use of Low Hamming Weight Product (LHWP) to compute a random power in a group or a multiple of an element in a ring. It reduces the computation of powers in a group with fast endomorphisms such as the Galois field F2n and Koblitz elliptic curves. In this paper, we introduce a reduced representation of LHWP and apply them to attack the relevant cryptosystems.

In this paper the problem for the improvement of the Delsarte bound for τ -designs in finite polynomial metric spaces is investigated. First we distinguish the two cases of the Hamming and Johnson Qpolynomial metric spaces and give exact intervals, when the Delsarte bound is possible to be improved. Secondly, we derive new bounds for these cases. Analytical forms of the extremal polynomials of ...

In this paper, we describe the properties of the i-components of Hamming codes. We suggest constructions of the admissible families of components of Hamming codes. It is shown that every q-ary code of length m and minimum distance 5 (for q = 3 the minimum distance is 3) can be embedded in a q-ary 1-perfect code of length n = (qm−1)/(q−1). It is also shown that every binary code of length m+k an...

We derive the Gilbert–Varshamov and Hamming bounds for packings of spheres (codes) in the Grassmann manifolds over and . Asymptotic expressions are obtained for the geodesic metric and projection Frobenius (chordal) metric on the manifold.

Abstract We study the structure of constacyclic codes of length ps over the ring Fpk + uFpk + · · · + um−1Fpk in terms of ideals in the quotient ring (Fpk + uFpk + · · · + um−1Fpk)[x]/〈xp s − λ〉, where λ is a unit element in Fpk + uFpk + · · · + um−1Fpk . In particular, we complete a structural classification of constacyclic codes of length ps over Fpk + uFpk and determine their minimum Hamming...

We consider vectors from {0, 1}. The weight of such a vector v is the sum of the coordinates of v. The distance ratio of a set L of vectors is dr(L) := max{ρ(x, y) : x, y ∈ L}/min{ρ(x, y) : x, y ∈ L, x 6= y}, where ρ(x, y) is the Hamming distance between x and y. We prove that (a) there are no positive constants α and C such that every set K of vectors with weight p contains a subset K′ with |K...