Improvements on the rates of iterated codes for error-free decoding on the binary symmetric channel are presented. Approximations to the performance of Elias's original error-free codes are replaced with virtually exact results that demonstrate higher code rates and the ability to decode from noisier channels than the original results indicated. Prefacing an Elias code with iterations of one or more primitive Bose-Chaudhuri-Hoequenghem (BCH) codes is shown to provide error-free decoding for any channel withp \leq 0.42and to yield code rates closer to capacity than those of Elias's original code. An heuristic algorithm is given for selecting an efficient set of BCH codes to iterate.

This paper studies the effectiveness of certain group codes in reducing the root-mean-square (rms) error in a digital telemetry link. The zero-sampling-error lower bound on the total error is derived for the case when majority-logic-decodable group codes are used in the channel. Calculated data are presented for several such codes including two new cyclic unequal-error-protection codes.

This correspondence presents a new class of cyclic majority-logic decodable codes. The codes provide unequal error protection for the information digits, i.e., some decoded digits are guaranteed to be correct despitet_1or fewer channel errors even though the minimum distance of the code guarantees protection from onlyt_0errors andt_1 > t_0.

This paper shows that, although systematic cyclic codes have equal error protection for all information digits, there exist non-systematic cyclic codes that can provide unequal error protection for the information digits; that is, at least one information digit is protected for a number of errors that is larger than the number guaranteed by the minimum distance of the code.

A recent paper on polynomial codes[l] presented an important theorem concerning the BCH minimum-distance bound for the dual of a polynomial code. For a particular set of numerical examples, however, the theorem failed. This correspondence presents a modified version of the theorem that covers those cases and uses a method of proof, which, while rigorous, shows clearly the significance of certain concepts. The BCH bound on the minimum distance, however, remains the same.

A new class of codes, called product generator codes, which are similar to Elias's iterated codes, are investigated. An important subclass of these codes is the generalized Reed-Muller codes. If the original codes that are iterated to produce a product code areL_1andL_2-step orthogonalizable, then the product code is (L_1 + L_2 - 1)-step orthogonalizable. Further if a\nuth-order product generator code is produced from these originalL-step orthogonal izable codes, the new code is at most(\nu +1)step orthogonalizable.

A necessary condition for the existence of a set ofJorthogonal parity check equations is developed, and it is demonstrated that, except for the trivial codes, the Reed-Solomon codes are notL-step orthogonalizable. Massey's concept of threshold decoding is generalized, and it is demonstrated that the Reed-Solomon codes are completely threshold decodable. Since every BCH code is a subcode of some Reed-Solomon code, every BCH code has a generalized threshold decoder.

This correspondence shows the formal equivalence between Massey's decoding scheme called threshold decoding involvingL-step orthogonalizable codes and Reed's decoding scheme originally conceived for the Muller codes. Upon examining these two decoding algorithms it is shown that each can be described in terms of a decoding logic circuit. The formal equivalence of the algorithms is proved by showing the formal equivalence of their respective decoding circuits.

It is proved that every linear code of dimensionkcan be decoded by a threshold decoding circuit that is guaranteed to correcteerrors ife \leq (d - 1)/2wheredis the minimum distance of the code. Moreover it is demonstrated that the number of levels of threshold logic is less than or equal tokby giving an algorithm for generating the decoding logic employingklevels.