Gérard D. Cohen

Further results on the covering radius of codes

A number of upper and lower bounds are obtained for K(n, R) , the minimal number of codewords in any binary code of length n and covering radius R . Several new constructions are used to derive the upper bounds, including an amalgamated direct sum construction for nonlinear codes. This construction works best when applied to normal codes, and we give some new and stronger conditions which imply that a linear code is normal. An upper bound is given for the density of a covering code over any alphabet, and it is shown that K(n + 2, R + 1) \leq K(n, R) holds for sufficiently large n .

Optimal Iris Fuzzy Sketches

Fuzzy sketches, introduced as a link between biometry and cryptography, are a way of handling biometric data matching as an error correction issue. We focus here on iris biometrics and look for the best error-correcting code in that respect. We show that two-dimensional iterative min-sum decoding leads to results near the theoretical limits. In particular, we experiment our techniques on the iris challenge evaluation (ICE) database and validate our findings.

Theoretical and Practical Boundaries of Binary Secure Sketches

Fuzzy commitment schemes, introduced as a link between biometrics and cryptography, are a way to handle biometric data matching as an error-correction issue. We focus here on finding the best error-correcting code with respect to a given database of biometric data. We propose a method that models discrepancies between biometric measurements as an erasure and error channel, and we estimate its capacity. We then show that two-dimensional iterative min-sum decoding of properly chosen product codes almost reaches the capacity of this channel. This leads to practical fuzzy commitment schemes that are close to theoretical limits. We test our techniques on public iris and fingerprint databases and validate our findings.

A Hypergraph Approach to the Identifying Parent Property: The Case of Multiple Parents

Let C be a code of length n over an alphabet of q letters. An n-word y is called a descendant of a set of t codewords x1, . . . ,xt if

Intersecting codes and independent families

y_i\in\{x^1_i,\dots,x^t_i\}

Error-correcting WOM-codes

for all i=1, . . . ,n. A code is said to have the t-identifying parent property if for any n-word that is a descendant of at most t parents it is possible to identify at least one of them. We prove that for any

On binary constructions of quantum codes

t\le q-1

A nonconstructive upper bound on covering radius

there exist sequences of such codes with asymptotically nonvanishing rate.

LINEAR INTERSECTING CODES

A binary intersecting code is a linear code with the property that any two nonzero codewords have intersecting supports. These codes appear in a wide variety of contexts and applications, e.g., multiple access, cryptography, and information theory. This paper is devoted partly to the study of intersecting codes, and partly to their use in constructing large t-independent families of binary vectors. The latter subject has by now been extensively studied and has application in VLSI testing, defect correction, E-biased probability spaces, and derandomization. By concatenation methods we construct codes with the highest known fate asymptotically. We then generalize the concept to t-wise intersecting codes: we give bounds on the achievable rate of such codes, both existential and constructive. We show how t-wise intersecting codes can be used to obtain (t+1)-independent families. With this method we obtain improved asymptotical constructions of t-independent families. Complexity issues are discussed. >

Bounds for Codes Identifying Vertices in the Hexagonal Grid

A problem raised by R.L. Rivest and A. Shamir (1982), namely, constructing write-once-memory (WOM) codes capable of error correction, is considered. The authors call a (n,m,t)-WOM code a scheme that allows t successive writings of m arbitrary bits (i.e., one message among 2/sup m/) on a WOM of size n. WOM codes have been studied from an information-theoretic viewpoint by J.K. Wolf et al. (1984) and constructed using classical coding theory by G.D. Cohen et al. (1986, 1987) (for example, with parameters, (23,11,3), (2/sup m-1/,m,2/sup m-2/+2/sup m-4/+1)). The authors adapt those methods in order to solve the problem raised by Rivest. Large classes of easily decodable single-error-correcting WOM codes are obtained. >