Testing a Code-Spotting Idea on More Code Types

The idea being tested

An earlier paper said it found a way to sort codes into two groups. The first group holds the best possible codes. The second group holds random junk codes. The test uses a math tool called the Mahler expansion. This tool turns a code's weight pattern into a set of numbers. Then it checks how many times two can divide into those numbers. The highest count across all numbers was called the top score. The earlier paper said best codes score twenty-eight. Random codes score four.

The earlier paper tested four code families. This paper tests eight more. Some tests use exact weight patterns from published books. Other tests use rough guesses where exact patterns are not known.

What was checked

The new tests looked at eight code types. Two came from a famous perfect code called the Golay code. One was a tiny five-bit code. One was a seven-bit code called Steane. One was a nine-bit code called Shor. Three were surface codes of different sizes. The Golay codes have exact weight patterns. The others do too, except the surface codes and the Shor code used rough guesses.

What the scores were

The two Golay-derived codes scored twenty-eight. This matched the earlier paper. But every other code scored between one and six. The random baseline was four. So the other codes looked just like random junk. The five-bit code scored one. The Steane code scored two. The Shor code scored two. The surface codes scored two, four, and six. None of these matched the claimed twenty-eight.

Why the trick fails

The Golay code has a very special shape. Its weight pattern mirrors itself perfectly. This extreme balance creates big scores in the math trick. Most codes lack this mirror shape. The earlier paper probably tested only Golay-type codes when it said "best codes." The label "best" was true for those codes. But the math trick does not work for best codes in general. It works only for this one mirror-shaped family.

What this means

The idea that the math trick spots best codes does not hold up. It survives only in a weaker form. The weaker form says mirror-shaped Golay-type codes get high scores. That is all. The big gap between twenty-eight and four is a fluke of one code family. It is not a universal rule. The paper leaves open whether any non-Golay code could ever score high. It also leaves open whether someone else can repeat the Golay result.

Words used here

Mahler expansion

a math tool that turns a pattern of weights into a new set of numbers, like reshuffling a deck of cards into a new order

Weight distribution

a count of how many pieces in a code have each possible weight, like counting how many people in a room wear each shoe size

Self-dual

a code whose weight pattern mirrors itself perfectly, like a face that looks the same on both sides

Stabilizer code

a type of quantum error-fixing code built from math rules that keep certain patterns steady

Learn more in plain language

• Quantum error correctionWikipedia (English, more technical)
• Binary Golay codeWikipedia (English, more technical)
• P-adic numberSimple English Wikipedia
• Coding theorySimple English Wikipedia
• Mahler's theoremWikipedia (English, more technical)
• Weight distributionWikipedia (English, more technical)