Let n; k; d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF (q). In this paper, eighteen codes are constructed which improve the known lower bounds on minimum distance.

We show that for any convex object Q in the plane, the average distance from the Fermat-Weber center of Q to the points in Q is at least ∆(P )/7, where ∆(P ) is the diameter of P , and that there exists a convex object for which this distance is ∆(P )/6. We use this result to obtain a linear-time approximation scheme for finding an approximate Fermat-Weber center of a convex polygon Q.

Let Γ denote a bipartite distance-regular graph with diameter D. In [J. S. Caughman, Bipartite Q-polynomial distance-regular graphs, Graphs Combin. 20 (2004), 47–57], Caughman showed that if D > 12, then Γ is Q-polynomial if and only if one of the following (i)-(iv) holds: (i) Γ is the ordinary 2D-cycle, (ii) Γ is the Hamming cube H(D, 2), (iii) Γ is the antipodal quotient of the Hamming cube H...

Let GF (q) denote the Galois field of q elements, and let V (n, q) denote the vector space of all ordered n-tuples over GF (q). The number of nonzero positions in a vector x ∈ V (n, q) is called the Hamming weight wt(x) of x. The Hamming distance d(x,y) between two vectors x,y ∈ V (n, q) is defined by d(x,y) = wt(x − y). A linear code C of length n and dimension k over GF (q) is a k-dimensional...

A q-ary maximum distance separable (MDS) code C with length n, dimension k over an alphabet A of size q is a set of qk codewords that are elements of An, such that the Hamming distance between two distinct codewords in C is at least n− k+1. Sets of mutually orthogonal Latin squares of orders q ≤ 9, corresponding to two-dimensional q-ary MDS codes, and q-ary one-error-correcting MDS codes for q ...

We find lower bounds on the minimum distance and characterize codewords of small weight in low-density parity check codes defined by (dual) classical generalized quadrangles. We analyze the geometry of the non-singular parabolic quadric in PG(4, q) to find information about the low-density parity check codes defined by Q(4, q), W(q) and H(3, q). ForW(q) and H(3, q), we are able to describe smal...

Clarkson recently introduced the o-smoothed distance do(p,q) = 2d(p,q)/(d(o, p)+d(o,q)+d(p,q)) (where d denotes the Euclidean distance in the plane) as a geometric analogue of the Jaccard distance; its Voronoi diagrams can be used to determine for a query point q the site p maximizing the dilation (d(p,o)+d(o,q))/d(p,q) of p and q in a star network centered at o. Although smoothed distance is n...

We present an algorithm that achieves general syndrome decoding of a (n; k; r) linear rank distance code over GF(q m) in O((nr + m) 3 q (m?r)(r?1)) elementary operations. As a consequence, the cryptographic Al schemes Che94, Che96] which rely on this problem are not secure with the proposed parameters. We also derive from our algorithm a bound on the minimal rank distance of a linear code which...

For two given simple polygons P Q the problem is to determine a rigid motion I of Q giving the best possible match between P and Q i e minimizing the Hausdor distance between P and I Q Faster algorithms as the one for the general problem are obtained for special cases namely that I is restricted to translations or even to translations only in one speci ed direction It turns out that determining...

In this paper we present a method to search q circulant matrices; the concatenation of these circulant matrices with circulant identity matrix generates quasi-cyclic codes with high various code rate q/(q+1) (q an integer). This method searches circulant matrices in order to find the good quasi-cyclic code (QCC) having the largest minimum distance. A modified simulated annealing algorithm is us...