Random Matrices and Codes for the Erasure Channel

Abstract

The design of erasure correcting codes and their decoding algorithms is now at the point where capacity achieving codes are available with decoding algorithms that have complexity that is linear in the number of information symbols. One aspect of these codes is that the overhead (number of coded symbols beyond the number of information symbols required to achieve decoding completion with high probability) is linear in k. This work considers a new class of random codes which have the following advantages: (i) the overhead is constant (in the range of 5 to 10), independent of the number of data symbols being encoded (ii) the probability of completing decoding for such an overhead is essentially one (iii) the codes are effective for a number of information symbols as low as a few tens (iv) the only probability distribution required is the uniform distribution. The price for these properties is that the decoding complexity is greater, on the order of k 3/2 . However, for the lower values of k where these codes are of particular interest, this increase in complexity might be outweighed by their advantages. The parity check matrices of these codes are chosen at random as windowed matrices, i.e. for each column an initial starting position of a window of length w is chosen and the succeeding w positions are chosen at random as zero or one. It can be shown that it is necessary that w = O(k 1/2 ) for the probabilistic matrix rank properties to behave as a non-windowed random matrix. The sufficiency of the condition has so far been established by extensive simulation, although other arguments strongly support this conclusion. The properties of the codes described 606 Algorithmica (2010) 56: 605-620 depend heavily on the rank properties of random matrices over finite fields. Known results on such matrices are briefly reviewed and several conjectures needed in the discussion of the code properties, are stated. The likelihood of the validity of the conjectures is supported through extensive experimentation. Mathematical proof of the conjectures would be of great value for their own interest as well of the particular coding application described here.

Adaptive Error-Correction Coding Scheme for Underwater Acoustic Communication Networks

“…For the case of a single packet transmission (denoted above as "single"), we demonstrate the gain of the adaptive coding scheme by showing performance of different coding implementations: fixed RS (F-RS) and punctured RS (P-RS) with coded word of 8 b, Fountain [24], and Raptor codes using an RS outer code [we note that similar values were obtained for Raptor codes using a low-density parity check (LDPC) outer code]. As performance of the RS scheme varies greatly with the erasure patterns, we show performance results for both independent identically distributed (i.i.d.)…”

“…However, since in underwater acoustic communication, transmission rate is on the order of a few kilobits per second and packets are small [23], we expect to be on the order of a few tens to thousand symbols. Therefore, since popular LT and Raptor codes perform well only for large code word lengths, for our numerical results, we apply the Fountain coding scheme described in [24], where good performance results are obtained for information word lengths as low as , at the cost of somewhat increased decoding complexity. When the channel cannot be modeled as an erasure channel, the integration of newly arrived samples into message-passing decoding is somewhat more complicated.…”

Comments on Matrix-Based Secret Sharing Scheme for Images

“…A.3.3) that a real-valued matrix filled with independent and identically distributed random variables, with continuous probability distribution function, will be singular with probability zero. On the contrary, in GF (2) the probability a square random binary matrix is singular as its dimension tends to infinity is 71.1 [15]. The scheme proposed in [13] stated that the random matrices A and B must be singular without specifying the value of the ranks.…”

On a class of optimal rateless codes

“…The performance issues described above lead to the question of whether it is possible to get better intermediate decoding performance, and hence better delay performance, by changing some of the constraints. In Studholme and Blake [11] it was shown that by changing the class of degree distributions allowed and the decoding algorithm, it is possible to decode all the uncoded packets with almost constant overhead irrespective of the block length. In this paper we complement this result by showing, with a modification of a known code construction, that changing the class of degree distributions allowed and the decoding algorithm, allows one to also achieve optimal intermediate decoding universally, i.e., for any given r ∈ [0, 1], (asymptotically) the decoder does not need to receive a linear excess of coded packets over rn .…”