An automated theorem proving technique is developed and is used to show that the capacity of the hexagonal (d, k) constraint is zero whenever k = d + 3 for d = 3,4, 5,7,9,11

Two sufficient conditions are given for the existence of binary fix-free codes (i.e., both prefix-free and suffix-free). Let L be a finite multiset of positive integers whose Kraft sum is at most 3/4. It is shown that there exists a fix-free code whose codeword lengths are the elements of L if either of the following two conditions holds: i) The smallest integer in L is at least 2, and no integer in L, except possibly the largest one, occurs more than 2/sup min(L)-2/ times. ii) No integer in L, except possibly the largest one, occurs more than twice. The results move closer to the Ahlswede-Balkenhol-Khachatrian conjecture that Kraft sums of at most 3/4 suffice for the existence of fix-free codes.

Binary prefix-free codes in which all codewords end with a "1" have been introduced by Berger and Yeung (1990). A recursive method is given here for the construction of all optimal "1"-ended codes with n codewords. It is shown that the set of codes obtained by the construction contains only optimal codes. We also compute recursively the number of essentially different optimal "1"-ended codes with n codewords and show that their number grows faster than any polynomial in n.

One of the key identities in linear cryptanalysis is the piling-up lemma, which allows one to compute the probability distribution of a sum modulo 2 of binary random variables, when the probability that these are zero is known. However, the lemma holds only for independent random variables. In linear cryptanalysis, one often (mis)uses this identity without knowing whether the random variables are independent or not. This paper investigates the problems that may arise when using this identity for dependent random variables. In particular, it is shown that the identity holds in almost all cases if one replaces equality by an approximation sign. The amplitude of departure from equality is also given.