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Shortest self-orthogonal embeddings of binary linear codes
Information Theory · Submitted 7 Nov 2025
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this paper, we use properties of the hull of a linear code to determine the length of a shortest self-orthogonal embedding of any binary linear code.
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The abstract, sentence by sentence
There has been recent interest in the study of shortest self-orthogonal embeddings of binary linear codes, since many such codes are optimal self-orthogonal codes. Several authors have studied the length of a shortest self-orthogonal embedding of a given binary code $\mathcal C$, or equivalently, the minimum number of columns that must be added to a generator matrix of $\mathcal C$ to form a generator matrix of a self-orthogonal code. In this paper, we use properties of the hull of a linear code to determine the length of a shortest self-orthogonal embedding of any binary linear code. We focus on the examples of Hamming codes and Reed-Muller codes. We show that a shortest self-orthogonal embedding of a binary Hamming code is self-dual, and propose two algorithms to construct self-dual codes from Hamming codes $\mathcal H_r$. Using these algorithms, we construct a self-dual $[22, 11, 6]$ code, called the shortened Golay code, from the binary $[15, 11, 3]$ Hamming code $\mathcal H_4$, and construct a self-dual $[52, 26, 8]$ code from the binary $[31, 26, 3]$ Hamming code $\mathcal H_5$. We use shortest SO embeddings of linear codes to obtain many inequivalent optimal self-orthogonal codes of dimension $7$ and $8$ for several lengths. Four of the codes of dimension $8$ that we construct are codes with new parameters such as $[91, 8, 42],\, [98, 8, 46],\,[114, 8, 54]$, and $[191, 8, 94]$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | There has been recent interest in the study of shortest self-orthogonal embeddings of binary linear codes, since many such codes are optimal self-orthogonal codes. | background 1.00 | 0.03 | 0.02 | asserted 0.99 |
| s1 | Several authors have studied the length of a shortest self-orthogonal embedding of a given binary code $\mathcal C$, or equivalently, the minimum number of columns that must be added to a generator matrix of $\mathcal C$ to form a generator matrix of a self-orthogonal code. | background 1.00 | 0.04 | 0.03 | asserted 0.99 |
| s2 | In this paper, we use properties of the hull of a linear code to determine the length of a shortest self-orthogonal embedding of any binary linear code. | method 0.54 | 0.96 | 0.51 | asserted 1.00 |
| s3 | We focus on the examples of Hamming codes and Reed-Muller codes. | contribution 0.85 | 0.71 | 0.03 | asserted 0.83 |
| s4 | We show that a shortest self-orthogonal embedding of a binary Hamming code is self-dual, and propose two algorithms to construct self-dual codes from Hamming codes $\mathcal H_r$. | result 0.73 | 0.96 | 0.32 | asserted 1.00 |
| s5 | Using these algorithms, we construct a self-dual $[22, 11, 6]$ code, called the shortened Golay code, from the binary $[15, 11, 3]$ Hamming code $\mathcal H_4$, and construct a self-dual $[52, 26, 8]$ code from the binary $[31, 26, 3]$ Hamming code $\mathcal H_5$. | result 0.95 | 0.97 | 0.15 | asserted 1.00 |
| s6 | We use shortest SO embeddings of linear codes to obtain many inequivalent optimal self-orthogonal codes of dimension $7$ and $8$ for several lengths. | result 0.53 | 0.96 | 0.20 | asserted 1.00 |
| s7 | Four of the codes of dimension $8$ that we construct are codes with new parameters such as $[91, 8, 42],\, [98, 8, 46],\,[114, 8, 54]$, and $[191, 8, 94]$. | result 0.98 | 0.96 | 0.17 | asserted 1.00 |
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The paper
Original abstract
There has been recent interest in the study of shortest self-orthogonal embeddings of binary linear codes, since many such codes are optimal self-orthogonal codes. Several authors have studied the length of a shortest self-orthogonal embedding of a given binary code $\mathcal C$, or equivalently, the minimum number of columns that must be added to a generator matrix of $\mathcal C$ to form a generator matrix of a self-orthogonal code. In this paper, we use properties of the hull of a linear code to determine the length of a shortest self-orthogonal embedding of any binary linear code. We focus on the examples of Hamming codes and Reed-Muller codes. We show that a shortest self-orthogonal embedding of a binary Hamming code is self-dual, and propose two algorithms to construct self-dual codes from Hamming codes $\mathcal H_r$. Using these algorithms, we construct a self-dual $[22, 11, 6]$ code, called the shortened Golay code, from the binary $[15, 11, 3]$ Hamming code $\mathcal H_4$, and construct a self-dual $[52, 26, 8]$ code from the binary $[31, 26, 3]$ Hamming code $\mathcal H_5$. We use shortest SO embeddings of linear codes to obtain many inequivalent optimal self-orthogonal codes of dimension $7$ and $8$ for several lengths. Four of the codes of dimension $8$ that we construct are codes with new parameters such as $[91, 8, 42],\, [98, 8, 46],\,[114, 8, 54]$, and $[191, 8, 94]$.
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References
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4 arXiv papers cite this in 7 sentences
Papers citing it · 5 sentences without a confident label
Cites it as background 2 papers
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Detailed information on shortest self-orthogonal embeddings can be found in [ref] .
Embedding linear codes over Z4 into self-orthogonal codes · Preliminaries · 0.93
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For shortest self-orthogonal and shortest LCD embeddings of linear codes over finite fields, we refer the reader to [ref] .
Shortest self-orthogonal and LCD embeddings of linear codes over Fq+uFq · Preliminaries · 0.81
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