LOCO Codes: Lexicographically-Ordered Constrained Codes

Abstract

Line codes make it possible to mitigate interference, to prevent short pulses, and to generate streams of bipolar signals with no direct-current (DC) power content through balancing. They find application in magnetic recording (MR) devices, in Flash devices, in optical recording devices, and in some computer standards. This paper introduces a new family of fixed-length, binary constrained codes, named lexicographically-ordered constrained codes (LOCO codes), for bipolar non-return-to-zero signaling. LOCO codes are capacity-achieving, the lexicographic indexing enables simple, practical encoding and decoding, and this simplicity is demonstrated through analysis of circuit complexity. LOCO codes are easy to balance, and their inherent symmetry minimizes the rate loss with respect to unbalanced codes having the same constraints. Furthermore, LOCO codes that forbid certain patterns can be used to alleviate inter-symbol interference in MR systems and inter-cell interference in Flash systems. Numerical results demonstrate a gain of up to 10% in rate achieved by LOCO codes with respect to other practical constrained codes, including run-length-limited codes, designed for the same purpose. Simulation results suggest that it is possible to achieve a channel density gain of about 20% in MR systems by using a LOCO code to encode only the parity bits, limiting the rate loss, of a low-density parity-check code before writing.

Asymmetric LOCO Codes: Constrained Codes for Flash Memories

“…For symmetric LOCO codes, it was shown in [15] that upon deciding the bridging method, there is a compromise between the maximum protection of the bits at the codeword transitions and the minimum number of bits/symbols to be used for bridging. The optimal bridging method for a symmetric LOCO code with parameter x in terms of bits protection was shown to require 2x bridging bits, which results in significant rate loss [15]. Thus, we adopted a suboptimal bridging method for symmetric LOCO codes that is bridging by x no writing, or no transmission, symbols.…”

“…Next, we discuss the self-clocking of A-LOCO codes. This feature is required in order to have clock recovery and system calibration [2], [15]. A constrained code is said to be selfclocked if any stream of codewords to be written contains a sufficient number of appropriately-separated transitions after bridging and signaling are applied.…”

“…In fact, for the larger length of 244 bits, the CA-LOCO code is within only 0.5% from the capacity. [15,Lemma 3]. Thus, balancing A-LOCO codes incurs a higher rate penalty.…”

Read-and-Run Constrained Coding for Modern Flash Devices

“…Because of their ability to improve performance via eliminating error-prone data patterns and undesirable sequences, constrained codes have a plethora of applications. They find application in one-dimensional (1D) magnetic recording devices, both the old ones, which are based on peak detection, and the modern ones, which are based on sequence detection [7], [8]. They can also be combined with robust signal detection using machine learning [9].…”

“…A number of recent results revisited [2] and [3] in order to produce efficient constrained codes based on lexicographic indexing, and one example is [19]. Another example is [7], in which we introduced binary symmetric lexicographicallyordered constrained (S-LOCO) codes and demonstrated density gains in a modern magnetic recording system. We extended our result to single-level cell (SLC) Flash (q = 2) [20] then to Flash with any number q of levels per cell [18].…”