Core Concepts
There are no perfect 2-error correcting codes over non-prime power alphabets for more than 170 new values of the alphabet size.
Abstract
The paper investigates the classification of perfect error-correcting codes over non-prime power alphabets, which has been an open problem for almost 50 years. The author focuses on the case of perfect 2-error correcting codes.
Key highlights:
- The author reduces the Hamming bound for perfect 2-error correcting codes to a Ramanujan-Nagell type Diophantine equation.
- By solving these Diophantine equations using techniques from computational number theory, the author proves the non-existence of perfect 2-error correcting codes over q-ary alphabets for more than 170 new values of q.
- The author also shows that there can only be finitely many perfect 2-error correcting codes over a non-prime power alphabet, consistent with the conjecture that such codes do not exist.
- The main technical challenges arise from the difficulty of determining the Mordell-Weil basis for certain elliptic curves, which is required to solve the Diophantine equations.
- The author discusses the limitations of the presented methodology and suggests potential future directions, such as extending the results to perfect quantum codes.
Stats
The paper provides the following key figures:
There are no perfect 2-error correcting codes over alphabets of size q ≤ 200, except for q = 94 and 166.
There are no perfect 2-error correcting codes over alphabets of size q ≤ 600, where all prime divisors of q are contained in the set {2, 3, 5, 7, 11}.
The paper lists the specific parameters (n, M) for perfect 2-error correcting codes over alphabets of size q = 15, 21, and 46.
Quotes
"The classification of perfect codes over non-prime power alphabets has been an open problem for which there have been no new results in almost 50 years."
"While this question has been completely settled if e ≥ 3, non-existence results are much more scarce if e ≤ 2."