Run length limited error control codes construction based on one control matrix property

Abstract

In this manuscript a simple method is presented for constructing run length limited error control codes from linear binary block codes. The run length limited properties are obtained via addition of a carefully chosen fixed binary vector - a modifier to all codewords without introducing any additional redundancy. Modifier selection is based on a specific property, which can be found in some of the linear binary block codes control matrices. Similar known methods are based on properties of generator matrices. However some codes are specified via control matrices, for example low density parity check codes. The method proposed in this letter could be applied to some of them directly. This is illustrated in this manuscript using example in which a run length limited low density parity check code is obtained from Gallager code.

On run-length limited error control codes constructed from binary product codes

“…Let us present a simple example from [5]. The wellknown binary [7,4,3] Hamming code has the following H matrix (6).…”

“…Such codes are denoted in literature as Run-Length Limited Error Control Codes (RLL-ECCs). A short overview of RLL-ECCs development and the current state of the art in their construction could be found in recent papers [4,5], which also form the main background knowledge useful for understanding this short communication.…”

Construction for obtaining trellis run length limited error control codes from convolutional codes

“…For example the most advanced codes used nowadays in many standards namely the LDPC codes and c-LDPC codes are defined by their control matrices H. It means that the methods based on generator matrices properties observed in [7] could not be applied directly to such codes. In [19] another property was proposed as an alternative, which could be observed in many H matrices in binary LBSECCs. Namely, if an even number of ones is discovered in a row of a control matrix in a set of positions ℜ, then negating an odd number of corresponding positions in a codeword will cause these positions to not all be the same.…”

“…On the other hand it would be appropriate to choose the intervals covering symbols participating in control equations as short as possible and, if possible, without any gaps between them in order to get minimal lengths of runs with identical symbols. In the matrix depicted in Fig 3 consecutive quadruples of ones are in rows (given in order corresponding to a sliding window with 20 columns): 1,17,18,19,20,16,32,33,34,35,31,47,48,49,50. These observations and requirements should be considered in the approach to construct an appropriate modifier.…”

“…The method was later improved gradually in [16][17][18] up to the stage that the error control deterioration caused by bit flipping was eliminated on the expense of quite complicated decoding techniques. Recently new construction method for obtaining RLL-ECCs from linear binary block ECCs was proposed in [19]. In this short communication it is shown that this method could be adapted also for obtaining trellis t-RLL-ECCs from convolutional ECCs (c-ECC).…”