# LOCO Codes: Lexicographically-Ordered Constrained Codes

## TL;DR

LOCO codes are a new family of capacity-achieving, lexicographically-ordered constrained codes for bipolar signaling, offering simple encoding/decoding, ease of balancing, and improved rate performance in magnetic and optical recording systems.

## Contribution

This paper introduces LOCO codes, a novel class of fixed-length, binary constrained codes with practical encoding/decoding and minimized rate loss, applicable to magnetic and Flash storage systems.

## Key findings

- LOCO codes achieve up to 10% higher rate than existing codes.
- LOCO codes enable about 20% channel density gain in magnetic recording.
- LOCO codes simplify encoding and decoding processes.

## 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.