On Redundancy Reduction of Non-Recursive Second-Order Spectral-Null Codes

The code design problem of non-recursive second-Order Spectral Null (2-OSN) codes is to convert balanced information words into 2-OSN words employing the minimum possible redundancy. Let $k$ be the balanced information word length. If $k\in \,\,2 {{I}}{{N}}$ then the 2-OSN coding scheme has length $n=k+r$ , with 2-OSN redundancy $r \in \,\,2 {{I}}{{N}}$ and $n\in \,\,4 {{I}}{{N}}$ . Here, we use a scheme with $r=2 \log k+\Theta (\log \log k)$ . The challenge is to reduce redundancy even further for any given $k$ . The idea is to exploit the degree of freedom to select from more than one possible 2-OSN encoding of a given balanced information word. To reduce redundancy, empirical results suggest that extra information $\delta _{k}=0.5 \log k+\Theta (\log \log k)$ is obtained. Thus, the proposed approach would give a smaller redundancy $r^{\prime }=1.5 \log k+\Theta (\log \log k)$ less than $r=2 \log k+\Theta (\log \log k)$

Capacity-approaching non-binary balanced codes using auxiliary data

It is known that, for large user word lengths, the auxiliary data can be used to recover most of the redundancy losses of Knuth’s simple balancing method compared with the optimal redundancy of balanced codes for the binary case. Here, this important result is extended in a number of ways. First, an upper bound for the amount of auxiliary data is derived that is valid for all codeword lengths. This result is primarily of theoretical interest, as it defines the probability distribution of the number of balancing indices that results in optimal redundancy. This result is equally valid for particular non-binary generalizations of Knuth’s balancing method. Second, an asymptotically exact expression for the amount of auxiliary data for the ternary case of a variable length realization of the modified balanced code construction is derived, that, in all respects, is the analogue of the result obtained for the binary case. The derivation is based on a generalization of the binary random walk to the ternary case and a simple modification of an existing generalization of Knuth’s method for the non-binary balanced codes. Finally, a conjecture is proposed regarding the probability distribution of the number of balancing indices for any alphabet size.The National Research Foundation (NRF) and SENTECH Chair in Broadband Wireless Multimedia Communication.http://ieeexplore.ieee.org/servlet/opac?punumber=18hj2019Electrical, Electronic and Computer Engineerin

Variable Length Unordered Codes

In an unordered code, no code word is contained in any other code word. Unordered codes are all unidirectional error detecting (AUED) codes. In the binary case, it is well known that among all systematic codes with k information bits, Berger codes are optimal unordered codes with r=[log2(k+1)] ≅ log2k check bits. This paper gives some new theory on variable length unordered codes and introduces a new class of systematic (instantaneous) unordered codes with variable length check symbols. The average redundancy of the new codes presented here is r ≅ (1/2)log2k+c, where c ∈ (1.0470,1.1332) ⊆ IR and k ∈ IN is the number of information bits. When k is large, it is shown that such redundancy is at most 0.6069 bits off the redundancy of an optimal systematic unordered code design with fixed length information symbols and variable length check symbols; and, at most 2.8075 bits off the redundancy of an optimal variable length unordered code design. The generalization is also given for the nonbinary case and it is shown that similar results hold true. keywords: {}, URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6145506&isnumber=614546

On systematic variable length unordered codes

In an unordered code no codeword is contained in any other codeword. Unordered codes are all unidirectional error detecting (AUED) codes. In the binary case, it is well known that among all systematic codes with k information bits, Berger codes are optimal unordered codes with r = 0.5log2(k+1)C check bits. This paper gives some new theory on variable length unordered codes and introduces a new class of systematic unordered codes with variable length check symbols. The average redundancy of these new codes is r = (1/2) log2 k + 1.047, where k is the number of information bits. It is also shown that such codes are optimal in the class of systematic unordered codes with fixed length information symbols and variable length check symbols. The generalization to the non-binary case is also given

Variable lenght unordered codes

In an unordered code no codeword is contained in any other codeword. Unordered codes are all unidirectional error detecting (AUED) codes. In the binary case, it is well known that among all systematic codes with k information bits, Berger codes are optimal unordered codes. This paper gives some new theory on variable length unordered codes and introduces a new class of systematic unordered codes with variable length check symbols. The average redundancy of these new codes is almost half one of Berger codes. It is also shown that such codes are optimal in the class of systematic unordered codes with fixed length information symbols and variable length check symbols. The generalization to the non-binary case is also given