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Abstract: An improved syndrome shift-register decoding algorithm, called the syndrome-weight decoding algorithm, is proposed for decoding three possible errors and detecting four errors in the (24,12,8) Golay code. This method can also be extended to decode two other short codes, such as the (15,5,7) cyclic code and the (31,16,7) quadratic residue (QR) code. The proposed decoding algorithm makes use of the properties of cyclic codes, the weight of syndrome, and the syndrome decoder with a reduced-size lookup table (RSLT) in order to reduce the number of syndromes and their corresponding coset leaders. This approach results in a significant reduction in the memory requirement for the lookup table, thereby yielding a faster decoding algorithm. Simulation results show that the decoding speed of the proposed algorithm is approximately 3.6 times faster than that of the algebraic decoding algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, an algebraic decoding algorithm is proposed to correct all patterns of four or fewer errors in the binary (41, 21, 9) Quadratic Residue (QR) code. The technique needed here to decode the (41, 21, 9) QR code is different from the algorithms developed in [I.S. Reed, T.K. Truong, X. Chen, X. Yin, The algebraic decoding of the (41, 21, 9) Quadratic Residue code, IEEE Transactions on Information Theory 38 (1992 ) 974–986]. This proposed algorithm does not require to solve certain quadratic, cubic, and quartic equations and does not need to use any memory to store the five large tables of the fundamental parameters in GF(220) to decode this QR code. By the modification of the technique developed in [R. He, I.S. Reed, T.K. Truong, X. Chen, Decoding the (47, 24, 11) Quadratic Residue code, IEEE Transactions on Information Theory 47 (2001) 1181–1186], one can express the unknown syndromes as functions of the known syndromes. With the appearance of known syndromes, one can solve Newton’s identities to obtain the coefficients of the error-locator polynomials. Besides, the conditions for different number of errors of the received words will be derived. Computer simulations show that the proposed decoding algorithm requires about 22% less execution time than the syndrome decoding algorithm. Therefore, this proposed decoding scheme developed here is more efficient to implement and can shorten the decoding time. [Copyright &y& Elsevier]

Nucleotide affinities for noncovalent interactions with amino acids produce associations between mRNAs and cognate peptides, potentially regulating ribosomal translation. Correlations between nucleotide affinities and residue hydrophobicity are explored for 25 theoretical minimal RNA rings, 22 nucleotide-long RNAs designed in silico to code for each amino acid once after three translation rounds, and forming stem-loop hairpins. This design presumably mimicks life's first RNAs. RNA rings resemble consensual tRNAs, suggesting proto-tRNA function, predicted anticodon and cognate amino acid. The 25 RNA rings and their presumed evolutionary order, deduced from the genetic code integration order of the amino acid cognate to their predicted anticodon, produces noteworthy associations with several ancient properties of the cell's translational machinery. Here we use this system to explore the evolution of codon affinity-residue hydrophobicity correlations, assuming these reflect pre-tRNA and pre-ribosomal translations. This hypothesis expects that correlations decrease with genetic code inclusion orders of RNA ring cognates. RNA ring associations between nucleotide affinities and residue hydrophobicities resemble those from modern natural genes/proteins. Association strengths decrease with genetic code inclusion ranks of proto-tRNA cognate amino acids. In silico design of minimal RNA rings didn't account for affinities between RNA and peptides coded by these RNAs. Yet, interactions between RNA rings and translated cognate peptides resemble modern natural genes. This property is strongest for ancient RNA rings, weakest for recent RNA rings, spanning a period during which modern tRNA- and ribosome-based translation presumably evolved. Results indicate that translation lacking tRNA-like adaptors based on codon-amino acid affinities and the genetic code pre-existed tRNA-mediated translation. Theoretical minimal RNA rings appear valid prebiotic peptide-RNA world models for the transition between pre-tRNA- and tRNA-mediated translations.