A construction based on Legendre sequences is presented for a doubly-extended binary linear code of length \(2p + 2\) and dimension \(p + 1\). This code has a double circulant structure. For \(p = 4k + 3\), we obtain a doubly-even self-dual code. Another construction is given for a class of triply extended rate \(1/3\) codes of length \(3p + 3\) and dimension \(p + 1\). For \(p = 4k + 1\), these codes are doubly-even self-orthogonal.
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