Preprint finds exact shortest code embeddings over finite rings

A new mathematical preprint gives exact answers to a specialized coding-theory problem: how many coordinates are needed to place a linear code over the ring Fq + uFq inside a longer code that is self-orthogonal or LCD. It gives formulas for the shortest length in each case and describes the constructions that attain those lengths.

An embedding here means retaining the original code while extending the construction with additional coordinates. The paper works with arbitrary linear codes over the ring, analyzing them through generator and Gram matrices. Its main outputs are theorem-level formulas for the minimum length and full descriptions of the shortest constructions.

The length depends on the field

The self-orthogonal result splits according to whether q is even or odd. For even q, the theorem gives one branch, $\mathrm{ns}(C) = n + r + 1$, under the paper’s stated residue, self-dot-product and rho conditions. In all other cases it gives $\mathrm{ns}(C) = n + r + \rho$. Here ns(C) is the length of a shortest self-orthogonal embedding, n is the original length, r is the rank over Fq of A0, the constant part of the Gram matrix, and rho is the rank over Fq of the specified restriction used in the theorem.

For odd q, the theorem first sets $m = \mu(-Z)$. The determinant branch then gives $\mathrm{ns}(C) = n + r + 2m$ or $\mathrm{ns}(C) = n + r + 2m + 1$. Here m is the quantity defined by the paper’s matrix-size function, Z is the lower-right block in the transformed A1 matrix, r is the rank of A0, and n is the original length. The choice between the two expressions is determined by the stated determinant branch.

The calculation begins by splitting a Gram matrix over the ring into two symmetric matrices over Fq, written as $A = A_0 + uA_1$. This reduces the embedding problem to congruence classification of symmetric and alternate matrices over finite fields. In ordinary language, that classification compares matrices by their essential form after changes of coordinates. The analysis also uses rank, radical, determinant and Witt-theory arguments.

The shortest cases are fully described

One theorem runs in the opposite direction. For a self-orthogonal code with positive free rank, every target length satisfying $n - k_1 \le n_0 < n$ can be realized as the length of a code whose shortest self-orthogonal embedding is the original code. Here n is the length of the original code, k1 is its free rank, and n0 is the length of the code being embedded.

Theorem 5.6 gives an if-and-only-if parametrization of all shortest self-orthogonal embeddings through a construction using matrices P, H and Q. The result therefore identifies the full family of minimum-length constructions, rather than providing only one way to reach the target length.

The LCD result is more compact: the exact shortest length is $n_{\ell}(C) = n + k - r$. In this formula, n is the original code length, k is the number of rows in the generator matrix, and r is the rank over Fq of A0. A companion theorem says every shortest LCD embedding has the stated appended-block form, with an invertible block satisfying $D \in \mathrm{GL}_{\ell}(R)$, where ell is the dimension of the radical of A0, and an otherwise unrestricted matrix E.

Examples make the formulas concrete

To show how the constructions work in specific cases, the authors used MAGMA to enumerate the shortest self-orthogonal and LCD embeddings. They then reported representatives with the largest minimum distance among the enumerated examples, applying Theorems 5.6 and 6.3.

Example 7.1 reports a shortest self-orthogonal embedding of length 27. Its calculation uses an original length of 20 and a rank contribution of 6 under the stated even-characteristic branch.

Example 7.2 also reports a shortest self-orthogonal embedding of length 27, calculated from an original length of 22, a rank of 5 and a radical-block rank rho of 0.

The next two self-orthogonal examples report lengths of 28 and 27. The first uses n equal to 23, rank r equal to 5 and m equal to 0; the second uses n equal to 22, r equal to 4, m equal to 0 and the additional branch offset of 1.

Example 7.5 shifts to the LCD problem and reports a shortest embedding length of 10, calculated from an original length of 8, three generator rows and a rank of 1 for A0.

A precise result with a narrow scope

The numerical examples need to be read at the level at which they are made. The MAGMA search covers the worked code sets, and the reported largest-minimum-distance representatives are tied to those enumerations. They do not by themselves establish a universal performance ranking for codes beyond the cases considered.

The conclusions concern exact lengths and algebraic constructions for linear codes in the stated ring setting. They are not measurements of an observed performance outcome or evidence from an empirical comparison.

The paper is a preprint on arXiv, identified as arXiv:2608.28222v1 and dated 28 Aug 2026.

Paper data and sources

Original title: Shortest self-orthogonal and LCD embeddings of linear codes over Fq+uFq
Authors: Junmin An, Jon-Lark Kim
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text