A Study on Error-Correcting Codes in {0,1}+

Concepts of channel-decodable codes were proposed in [3] and [4], which concern the requirements of the decodability and error-correctability of messages including errors caused by substitutions, deletions, or insertions. When a system is not synchronized, some segments of messages could be lost. Or, when we transmit messages through a noisy wireless channel, messages could be changed by deletions or insertions and no more keep the lengths of original messages. These errors can be modeled as symbol deletions or insertions. An easiest way in coding applications to detect the deletion or insertion errors is to use a code repeatedly to transmit a sequence of messages. But it is shown [1] that the coding scheme obtained by the repeated use of codes is bad, if used five times or more. Techniques to detect or to correct the bit insertion or deletion errors in messages coded by uniform codes have been studied in [6] and [7].As a general approach to investigate the decoding problems in message transmission, a model of abstract channels with insertion, deletion, or substitution errors was proposed in [3]. Some properties required for a code to be channel-decodable for a given abstract channel were studied in [3] and [4]. A further analysis of these abstract channels that involve combinations of three basic error types was given by Konstantinidis [5]. In [6], the error-detecting for finite length messages is studied. But for the error in infinite length messages (general case), the error detectable codes has not been investigated. Following this model of channels, some algebraic properties of channel-detectable and -correctable codes have been derived in [2]. A kind of channel-detectable and -correctable codes is also proposed in [2]. But more general properties of channel-correctable codes and maximal channel-correctable codes still have to be further investigated.[1] K. A. S. Abdel-Ghaffar, Repeated Use of Codes for Error Detection Five Times is Bad, IEEE Trans. Inform. Theory, Vol.48, No.7 (2002), 2053--2060.[2] H.K. Hsiao, Hsiu-Hsia Lin and S.S. Yu, The Decodability and Correctability of Codes, Int. J. Computer Math., (Accepted).[3] H. Jürgensen and S. Konstantinidis, Variable-Length Codes for Error Correction, Automata, languages and programming (Szeged, 1995), 581--592, Lecture Notes in Comput. Sci., 944, Springer, Berlin, 1995.[4] H. Jürgensen and S. Konstantinidis, Error Correction for Channels with Substitutions, Insertions, and Deletions, Information theory and applications, II (Lac Delage, PQ, 1995), 149--163, Lecture Notes in Comput. Sci., 1133, Springer, Berlin, 1996.[5] S. Konstantinidis, Structural Analysis of Error-Correcting Codes for Discrete Channels That Involve Combinations of Three Basic Error Types, IEEE Trans. Inform. Theory, Vol.45, No.1 (1999), 60--77.[6] S. Konstantinidis and Amber O'Hearn, Error-Detecting Properties of Languages, Theor. Comp. Sci., Vol.276 (2002), 355—375.[7] V. I. Levenshtein, Binary Codes Capable of Correcting Deletions, Insertions, and Reversals, Soviet Phys. Dokl., Vol.10, No.8 (1966), 707--710 (English translation).頻道可解碼的數碼模式概念是在[3], [4] 兩篇文章中所提出的，這包含資訊在具有替代、刪除或插入所產生錯誤的頻道中，要達成資訊的解碼與更正時之需求。在一個非同步的系統中，可能會有資料遺失。資訊在有雜訊的無線頻道中傳遞，資訊亦會有減少或增加的可能，而不復維持原資訊的長度。在偵測由刪除或插入所產生之錯誤時，一個最簡單的編碼方法是將資訊重覆的傳送。但在[1]一文中證明，重覆五次或更多次時，將資訊重覆傳送的編碼模式並不佳。運用同長度的數碼，來偵測由刪除或插入所產生之錯誤的技術，曾在[6], [7]兩篇文章中被研究到。在[3]一文中提出包含替代、刪除或插入所產生錯誤的抽象頻道模式，作為研究資訊傳輸的編碼問題之一般解決方案。一些關於數碼之抽象頻道的頻道可解碼性質，在[3], [4] 兩篇文章中曾被研究到。[5]中對混合三種基本錯誤型態的抽象頻道，曾作更進一步的分析。[6]中對有限長度序列之錯誤偵測，討論許多，但未討論無限長度序列之錯誤偵測。在此抽象頻道模式下，一些頻道可解碼與頻道可更正的數碼之代數性質，曾在[2]中被推導出來。在[2]中並提出一種頻道可解碼與可更正的數碼。然而更廣泛的頻道可更正數碼的性質，與更一般化的極大頻道可更正數碼，仍有待更進一步的研究與探討