Separation and Witnesses

Abstract

Abstract. We revisit two notions of difference between codewords, namely separation and the existence of small witnesses, and explore their links. IntroductionLet Q be an alphabet of size q. A subset C of Q n with |C| = M is an (n, M ) q or (n, M )-code. Elements c = (c 1 , . . . , c n ) of C are codewords. Let R = R(C) = log q M/n denote the rate of C.Coding theory asks for codes (or sets) C such that every codeword c ∈ C is as "different" as possible from all the others. The usual requirement is a large minimum Hamming distance between codewords; the associated question is to determine the maximum size of such a code.We survey here two relaxations of this problem, namely, the notions of separation and witness, and their interplay.In the first one, separation, we look for some minimum distance between disjoint subsets of codewords (instead of merely -singletons of-codewords).In the second relaxation, dealing with the existence of a witness, we look for a small subset W ⊂ [n] of coordinates such that c differs from every other codeword in W . In other words, c can be singled out from all the other codewords by observing only a small subset of coordinates.We then establish links between some separating and witness codes and conclude with a few open problems. SeparationAs an introductory illustration before the general case, consider hashing, central in Computer Science and Coding, see, e.g., [12] and its references.For a parameter t ≥ 2 a code C is called t-hashing if for any t distinct codewords c 1 , . . . , c t ∈ C there is a coordinate 1 ≤ i ≤ n such that all values cAn obvious necessary condition for the existence of a t-hashing family is q ≥ t; it turns out to be sufficient too (see [11], [22], [23], [29] for bounds on the rate of t-hashing families of growing length).An extension of hashing was introduced in [6].

Some new results on equivalency of collusion‐secure properties for Reed‐Solomon codes

“…The code classes defined above are known to satisfy the following relationships. Proposition 1.1 (see [9]) Let d be the minimum distance of a code C of length n. [2]) Let d be the minimum distance of a code C of length n.…”

“…Proposition 1 (see [3]): Let d be the minimum distance of a code C of length n. Then for C, [4]): Let d be the minimum distance of a code C of length n.I fd > n(1 − 1/(w 1 w 2 )), then C is a (w 1 , w 2 )-separating code.…”

On metric convexity, the discrete Hahn-Banach theorem, separating systems and sets of points forming only acute angles

“…In his presentation [Coh09] he only makes a very brief mention. He encouraged me to publish all the details by myself, including heuristics and partial results supporting the underlying conjectures.…”