# The Secret Arithmetic of Patterns: A General Method for Designing Constrained Codes Based on Lexicographic Indexing

arXiv: https://arxiv.org/abs/2010.10686 · HTML: https://mlsanity.com/paper/2010.10686/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Oct 2020.

## Built on by (most cited first, 1 of 1)

- [Learning to Reconfigure: Using Device Status to Select the Right Constrained Coding Scheme](https://mlsanity.com/paper/2512.21396/) (2025)

## Abstract (arXiv)

Constrained codes are used to prevent errors from occurring in various data storage and data transmission systems. They can help in increasing the storage density of magnetic storage devices, in managing the lifetime of electronic storage devices, and in increasing the reliability of data transmission over wires. We recently introduced families of lexicographically-ordered constrained (LOCO) codes. These codes achieve capacity with simple encoding and decoding, and they are easy to reconfigure. In this paper, we generalize our work on LOCO codes by presenting a systematic method that guides the code designer to build any constrained code based on lexicographic indexing once the finite set of data patterns to forbid is known. In particular, we connect the set of forbidden patterns directly to the cardinality of the code and to the rule that uncovers the index associated with a codeword. By doing that, we reveal the secret arithmetic of patterns, and make the code design significantly easier. We design optimal (rate-wise) constrained codes for the new two-dimensional magnetic recording (TDMR) technology. We show notable performance gains as a result of solely applying the new codes. Moreover, we show how near-optimal constrained codes be designed and used to further reduce complexity.