Formula omitted]-circular codes

Abstract

A message such as mRNA, which consists of continuous characters without separators (such as commas or spaces), can easily be decoded incorrectly if it is read in the wrong reading frame. One construct to theoretically avoid these reading frame errors is the class of block codes. However, the first hypothesis of Watson and Crick (1953) that block codes are used as a tool to avoid reading frame errors in coding sequences already failed because the four periodical codons AAA, CCC, GGG and UUU seem to play an important role in protein coding sequences. Even the class of circular codes later discovered by Arquès and Michel (1996) in coding sequences cannot contain a periodic codon. However, by incorporating the interpretation of the message into the robustness of the reading frame, the extension of circular codes to include periodic codons is theoretically possible. In this work, we introduce the new class of I-circular codes. Unlike circular codes, these codes allow frame shifts, but only if the decoded interpretation of the message is identical to the intended interpretation. In the following, the formal definition of I-circular codes is introduced and the maximum and the maximal size of I-circular codes are given based on the standard genetic code table. These numbers are calculated using a new graph-theoretic approach derived from the classical one for the class of circular codes. Furthermore, we show that all 216 maximum self-complementary C3-codes (see Fimmel et al., 2015) can be extended to larger I-circular codes. We present the increased code coverage of the 216 newly constructed I-circular codes based on the human coding sequences in chromosome 1. In the last section of this paper, we use the polarity of amino acids as an interpretation table to construct I-circular codes. In an optimization process, two maximum I-circular codes of length 30 are found.