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On the lengths of MDS codes with a two-transitive permutation automorphism group
Haihua Deng, Tao Feng, Andrey V. Vasil'ev
TL;DR
The paper asks whether the MDS conjecture’s length bound holds for codes with 2-transitive permutation automorphism groups. It combines field-extension arguments with modular-representation analysis of relevant group actions. The result is n ≤ q + 1 for 4 ≤ k ≤ q − 3, while complete classification remains open for affine-type groups.
Problem
The paper studies whether the MDS conjecture can be verified for MDS codes whose permutation automorphism group is 2-transitive, amid an unavailable complete classification of such codes.
Method
The proof uses field-extension preservation and modular-representation results to reduce and analyze the relevant 2-transitive group families.
Results
For 4 ≤ k ≤ q − 3, every MDS code with a 2-transitive permutation automorphism group satisfies n ≤ q + 1.
Takeaways & Limitations
The MDS conjecture is verified for this class of codes, while determining all codes with 2-transitive groups remains an open problem.
Takeaways & Limitations
The classification is more challenging for affine-type groups because the required modular-representation information is not generally available.
Abstract
from arXiv · showhide
Let $C$ be an $[n,k]_q$ maximum distance separable (MDS) code with $4\le k\le q-3$, and suppose that it has a $2$-transitive permutation automorphism group. In this paper we show that $n\le q+1$, so the MDS conjecture holds for this class of codes.
1. Introduction
The introduction frames the MDS conjecture as a central length bound and studies whether strong permutation symmetry enforces it. The paper proves the bound n ≤ q + 1 for MDS codes with 2-transitive permutation automorphism groups in the stated parameter range.
- An [n, k, d]q code is MDS when it attains the Singleton bound, d = n − k + 1.
- The MDS conjecture predicts n ≤ q + 1 for nontrivial codes, with the exception q even and k ∈ {3, q − 1}, where n ≤ q + 2.
- The paper initiates the study of MDS-code lengths under large transitive permutation groups, while a complete classification of 2-transitive cases remains unavailable.
- Known 2-transitive examples include affine constructions, translation-hyperoval and Segre-arc codes, and the hexacode extension.
- Theorem 1.3 proves that 4 ≤ k ≤ q − 3 and 2-transitive PAut(C) imply n ≤ q + 1.
- The proof reduces the problem using modular representations of 2-transitive groups and then treats the resulting infinite families of almost simple groups.
2. Preliminaries
The preliminaries establish the code, duality, automorphism-group, field-extension, and module-theoretic tools used to analyze 2-transitive MDS codes. They also record general length bounds and reduction principles for codes over subfields.
- PAut(C) = PAut(C⊥), and a nontrivial code is MDS exactly when its dual is MDS.
- The permutation automorphism group is 2-transitive when it acts transitively on ordered pairs of distinct coordinate positions.
- Permutation automorphisms admit matrix representatives satisfying PgM = MBg, linking coordinate permutations to invertible transformations of a generator matrix.
- Extending the field preserves an MDS code’s parameters and permutation automorphism group.
- Every nontrivial [n, k]q MDS code satisfies n ≤ q + min{k, n − k} − 1, and hence n ≤ 2q − 2.
- Frobenius-invariant submodules can be realized over a subfield, yielding subfield MDS codes with the same dimension and applicable length bounds.
3. Reduction to the reducible hearts
The proof fixes the MDS code and its 2-transitive automorphism group, then reduces the problem through permutation-module structure and the classification of finite 2-transitive groups. Affine groups give n ≤ q directly, while the remaining almost simple cases are narrowed to tables and eliminated using lemmas and explicit code checks.
- Setup: The proof sets G = PAut(C), lets Ω be the coordinate positions, and assumes k ≤ n/2 without loss of generality.The code is treated as a subspace over the algebraic closure, with G acting 2-transitively on Ω.
- Case elimination: The cases listed in Table 3.1 are excluded because Lemma 3.1 establishes n ≤ q + 1 for each corresponding stabilizer configuration.The table records the relevant (G, Gα, p) tuples and derives possible q values using Corollary 2.6.
- Computational checks: Explicit checks of remaining invariant codes show that none of the listed codes is MDS.The computations include cases for 2G2(3), M24, M23, M22, and M11.
- Permutation-module reduction: If the heart A is simple, the only G-submodules of the permutation module M are 0, J, A, and M.Since a nontrivial MDS code has min{k, n−k} ≥ 2, it cannot be one of these four submodules; therefore the heart is reducible.
- Affine case: Affine 2-transitive automorphism groups force n ≤ q.The affine socle acts regularly on Ω, and the contradiction n ≥ q + 1 is obtained by comparing n = p^a with q = p^f.
- Almost simple case: If n > q + 1, the automorphism group must be almost simple, and (G, Ω, q) must occur in Table 3.2.The argument uses reducibility of the heart and excludes cases covered by known MDS-conjecture results, including the PSL2(11) case.
4. Almost simple actions
The paper reduces almost simple 2-transitive automorphism groups to six candidate families and excludes each through module and codeword arguments, leaving only the q case for PSL2(5).
- Reduction: Theorem 4.1 reduces the almost simple case to analyzing candidate 2-transitive actions listed in Table 3.2.The proof assumes S = soc(G) is nonabelian simple and examines the six cases individually.
- Suzuki groups: A module-chain bound gives n ≤ 2p − 2, which contradicts the parameter bounds for Suzuki socles Sz(2^(2a+1)).The contradiction excludes S = Sz(2^(2a+1)) for a ≥ 1.
- Linear groups: For S = PSL_d(p^s), d ≥ 3, hyperplane-design submodules yield codewords whose weights violate the MDS constraints, excluding this family.Theorem 4.3 concludes that soc(G) cannot be PSL_d(p^s).
- Symplectic groups: For symplectic actions S = Sp_2m(2), the constructed submodule B_ε lies in C ∩ C^⊥, and resulting codewords produce a contradiction.This excludes Sp_2m(2) for m ≥ 3.
- Remaining families: The remaining families PSU_3(r) with r > 2 and 2G_2(3^(2a+1)) are excluded, while PSL2(r) forces r = 5 and C to be a [6,3,4] code.For q a power of 4, the surviving PSL2(5) case is the unique 4 hexacode up to equivalence.
- Conclusion: Together with the affine bound and the reduction from n > q + 1 to Table 3.2, these exclusions complete the proof of n ≤ q + 1.Theorem 4.1 handles the almost simple candidates, and Proposition 3.4 connects this analysis to the main theorem.
5. Conclusions
The paper verifies the MDS conjecture for codes with 2-transitive permutation automorphism groups, while leaving complete classification open, especially for affine-type groups.
- The paper verifies the MDS conjecture for [n,k]_q MDS codes with a 2-transitive permutation automorphism group.The conclusion states this verification for codes C ≤ F^Ω with G = PAut(C) ≤ Sym(Ω).
- For almost simple-type families, every such code is equivalent to a scalar extension of the hexacode from F4 to Fq.
- Determining all MDS codes with a 2-transitive permutation group remains open for affine-type groups.The paper notes that the required modular-representation information is not generally available for these groups.
- The main theorem initiates the study of MDS codes with highly transitive permutation groups.The authors identify weaker symmetry hypotheses, particularly cyclic codes, as directions for further verification of the MDS conjecture.