Integer Codes Correcting Double Errors and Triple-Adjacent Errors Within a Byte

Abstract

This paper presents a class of integer codes that are suitable for use in optical computer networks where the data is transmitted serially. The presented codes are constructed with the help of a computer and have three desirable properties. First, they use integer and lookup table operations, which makes them suitable for software implementation. Second, depending on the application requirements, the proposed codes can be used as lower rate error correction (EC) codes or as high-rate error detection (ED) codes. In the EC mode, which is suited for realtime applications, the receiver can correct all single and double errors as well as all triple-adjacent (TA) errors within one b-bit byte. On the other hand, if the integrity of data is of high importance, the receiver may operate in the ED mode. In that case, it is able to detect all quadruple errors, all double TA errors within one b-bit byte and all double TA errors within two b-bit bytes. Finally, it is important to note that the presented codes can be interleaved without delay and without using any additional hardware. Owing to this, it is possible to construct simple codes capable of detecting/correcting multiple TA and random errors.

Integer codes correcting single errors and detecting burst errors within two bytes

“…The error control procedure for the proposed codes is very similar to that from [9][10][11][12][13]. In short, after receiving corrupted packet (S ≠ 0), the decoder will look up the syndrome table (ST) to find the entry with the error correction data (Figure 2).…”

“…Definition [13] Let $${Z}_{{2}^b - 1}$$ = {0, 1,…, 2 b ‒ 2} be the ring of integers modulo 2 b ‒ 1 and let $$\smash{{B}_i\, = \sum\nolimits_{n = \mathop 0\limits }^{b - 1} {{a}_n \cdot {{\mathop {\mathop 2\limits }\limits }}^n}} $$ be the integer representation of a b ‐bit byte, where $${a}_n \in $$ {0, 1} and 1 ≤ i ≤ k . Then, the code defined as C ( b, k, c ), $$$$\begin{eqnarray} \hskip-10pt C(b,\, k,\, c) = \left\{ {x \in Z_{{2}^b - 1}^{k + 1}{:}\sum_{\,i = 1}^{\,k} {{C}_i \cdot {B}_{i}} \equiv {B}_{k{+ 1}}({\rm mod}\,{2}^b - {1})}\!\right\} \end{eqnarray}$$$$is an ( kb + b, kb ) integer code, where x = ( B 1 , B 2 , …, B k , B k +1 ) $$ \in Z_{{2}^b - 1}^{k + 1}$$ is the codeword vector, c = ( C 1 , C 2 , … , C k , 1) $$ \in Z_{{2}^b - 1}^{k + 1}$$ is the coefficient vector and B k +1 $$ \in Z_{{2}^b - 1}$$ is an integer. Definition [13] Let x = ( B 1 , B 2...…”

“…In that case, exactly | s 1 | relationships (Theorem 1) between the nonzero syndrome (element of the set s 1 ), error location ( i ) and error value ( E ) will be established. So, if the ST is sorted, the decoder will find the corresponding entry after n 1 table lookups and n 1 comparisons, where$$\,1 \le {n}_1 \le \lfloor {log_{\,2}| {{s}_1} |} \rfloor + 2$$ [13].…”

Integer codes correcting single errors and double adjacent errors

“…The first step in constructing any integer code is to define the integer values of the errors to be corrected. In our case, this step is simplified, since from [23] we already know that the integer value of a single error is equal to e i = ± 2 r , where 0 ≤ r ≤ b ‒ 1 and 1 ≤ i ≤ k + 1. When it comes to DA errors, we know that they can corrupt one or two b ‐bit bytes.…”

Low overhead optimal parity codes

“…But the promising future of electronics at nanoscale allows the devices get closer where the devices undergo radiation effects giving rise to multiple event upsets that induce soft errors in memories [20]- [23]. For embedded memories, the correction capability must be as high as possible within a single clock cycle to ensure at-speed testing [24]- [28]. This paper addresses the correction of maximum number of erroneous bits as fast as possible by using minimum number of redundant bits and maximum code rate.…”