For discrete memoryless multiple-access channels, we propose a general definition of variable length codes with a measure of the transmission rates at the receiver side. This gives a receiver perspective on the multiple-access channel coding problem and allows us to characterize the region of achievable rates when the receiver is able to decode each transmitted message at a different instant of...

Recently [1] we proposed a family of magic state distillation protocols that obtains asymptotic performance that is conjectured to be optimal. This family depends upon several codes, called “inner codes” and “outer codes.” In Ref. [1], some small examples of these codes were given as well as an analysis of codes in the asymptotic limit. Here, we analyze such protocols in an intermediate size re...

Baranyai’s theorem is a well-known theorem in the theory of hypergraphs. A corollary of this theorem says that one can partition the family of all u-subsets of an n-element set into ( n−1 u−1 ) sub-families such that each sub-family form a partition of the n-element set, where n is divisible by u. In this paper, we present a codingtheoretic application of Baranyai’s theorem (or equivalently, th...

Physical Unclonable Functions can be used for secure key generation in cryptographic applications. It is explained how methods from coding theory must be applied in order to ensure reliable key regeneration. Based on previous work, we show ways how to obtain better results with respect to error probability and codeword length. Also, an example based on Generalized Concatenated codes is given, w...

The unequal error correction capabilities of binary cyclic codes of composite length aye investigated. Under certain conditions, direct sums of concatenated codes have unequal error correction capabilities. By a modified Hartmann and Tzeng algorithm, it is shown that a binary cyclic code of composite length is equivalent to the direct sum of concatenated codes. With this, some binary cyclic une...

We describe a concatenated coding system with iterated sequential inner decoding. The system uses convolutional codes of very long constraint length and operates on iterations between an inner Fano decoder and an outer Reed-Solomon decoder.

Deletion correction codes have been used for transmission synchronization and, more recently, tracing pirated media. A generalized Reed-Solomon (GRS) code, denoted by GRSk(l,q,alpha,v), is a code of length l over GF(q) with qk codewords. These codes have an efficient decoding algorithm and have been widely used for error correction and detection. It was recently demonstrated that GRS codes are ...

The number of correctable errors of weight half the minimum distance is given for the first-order binary Reed-Muller codes. It is shown that the size of trial set, which is introduced by Helleseth et al. and can be used for a minimum distance decoding or estimating the number of uncorrectable errors, for the first-order Reed-Muller codes is at least that of minimal codewords except for small code

In error-correcting codes for combating noisy transmission channels, a central concept i s the notion of minimum distance. If a code can be constructed with minimum distance between code points of 2m-k 1 I then any number of errors per code word which does not exceed m can be corrected, thus increasing the reliability of transmission above that to be expected with no redundancy i n the code. An...

t immaru-A q-nary error-correcting code with N = qk code words of length n = k + r can have no greater miniium distance d than r + 1. The class of codes for which d = r + 1 is studied first in general, then with the restriction that the codes be linear. Examples and construction methods are given to show that these codes exist for a number of values of q, k, and r. Proof: Pick any k position. T...