# On the Maximum Number of Codewords of X-Codes of Constant Weight Three

## TL;DR

This paper establishes a lower bound on the maximum size of constant-weight X-codes used in circuit testing, demonstrating the existence of infinitely many such codes with increasing compaction ratios, and provides an efficient construction method.

## Contribution

It introduces the first nontrivial lower bound for the maximum number of codewords in constant-weight X-codes with specific parameters and offers a polynomial-time algorithm to construct codes meeting this bound.

## Key findings

- Existence of an infinite sequence of X-codes with unbounded compaction ratio for fixed d.
- A nontrivial lower bound on the maximum number of codewords for (m, n, d, 2) X-codes.
- A polynomial-time algorithm to construct X-codes achieving the bound.

## Abstract

X-codes form a special class of linear maps which were originally introduced for data compression in VLSI testing and are also known to give special parity-check matrices for linear codes suitable for error-erasure channels. In the context of circuit testing, an $(m, n, d, x)$ X-code compresses $n$-bit output data $R$ from the circuit under test into $m$ bits, while allowing for detecting the existence of an up to $d$-bit-wise anomaly in $R$ even if up to $x$ bits of the original uncompressed $R$ are unknowable to the tester. Using probabilistic combinatorics, we give a nontrivial lower bound for any $d \geq 2$ on the maximum number $n$ of codewords such that an $(m, n, d, 2)$ X-code of constant weight $3$ exists. This is the first result that shows the existence of an infinite sequence of X-codes whose compaction ratio tends to infinity for any fixed $d$ under severe weight restrictions. We also give a deterministic polynomial-time algorithm that produces X-codes that achieve our bound.