We give constructions of self-dual and formally self-dual codes from group rings where the ring is a finite commutative Frobenius ring. We improve the existing construction given in [11] by showing that one of the conditions given in the theorem is unnecessary and moreover it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes correspond to ideals in the group ring RG and as such must have an automorphism group that contains G as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative [72, 36,16] Type II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48.

We introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring F 2 + uF 2 , using groups of * corresponding author orders 9, 15, 21, 25, 27, 33 and 35 to find extremal binary self-dual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary self-dual codes of length 72. In particular we obtain 41 new binary extremal self-dual codes of length 68 from groups of orders 15 and 33 using neighboring and extensions. All the numerical results are tabulated throughout the paper.

<p style='text-indent:20px;'>In this paper, we construct new self-dual codes from a construction that involves a unique combination; <inline-formula><tex-math id="M1">\begin{document}$ 2 \times 2 $\end{document}</tex-math></inline-formula> block circulant matrices, group rings and a reverse circulant matrix. There are certain conditions, specified in this paper, where this new construction yields self-dual codes. The theory is supported by the construction of self-dual codes over the rings <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{F}_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{F}_2+u \mathbb{F}_2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{F}_4+u \mathbb{F}_4 $\end{document}</tex-math></inline-formula>. Using extensions and neighbours of codes, we construct <inline-formula><tex-math id="M5">\begin{document}$ 32 $\end{document}</tex-math></inline-formula> new self-dual codes of length <inline-formula><tex-math id="M6">\begin{document}$ 68 $\end{document}</tex-math></inline-formula>. We construct 48 new best known singly-even self-dual codes of length 96.</p>

In this work, we describe a double bordered construction of self-dual codes from group rings. We show that this construction is effective for groups of order 2p where p is odd, over the rings F 2 + uF 2 and F 4 + uF 4. We demonstrate the importance of this new construction by finding many new binary self-dual codes of lengths 64, 68 and 80; the new codes and their corresponding weight enumerators are listed in several tables. Keywords Group rings • Self-dual codes • Codes over rings • Extremal codes • Bordered constructions Mathematics Subject Classification (2010) 94B05 • 94B15 This research was supported by the London Mathematical Society (International Short Visits-Scheme 5).

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