关于等对偶双Toeplitz码
On isodual double Toeplitz codes
摘要：
双Toeplitz（简称DT）码是双循环码的推广。我们表明，这样的代码是isodual，因此正式自对偶。自对偶DT码的特征是双循环码或双双非循环码。同样地，甚至DT二进制码也被表征为双循环码。通过穷举搜索得到的数值例子表明，所构造的码在形式上的自对偶码中具有最好的已知最小距离，最大可达一个单位，并且有时在已知值上有所改进。在$\F_4$上，基于素数域中的二次剩余的DT码的显式构造同样表现良好。我们证明了DT码在$\F_q$上是渐近好的。具体来说，我们构造DT码任意接近的渐近varshamov-Gilbert界的代码率的一半。
英文摘要：
Double Toeplitz (shortly DT) codes are introduced here as a generalization of double circulant codes. We show that such a code is isodual, hence formally self-dual. Self-dual DT codes are characterized as double circulant or double negacirculant. Likewise, even DT binary codes are characterized as double circulants. Numerical examples obtained by exhaustive search show that the codes constructed have best-known minimum distance, up to one unit, amongst formally self-dual codes, and sometimes improve on the known values. Over $\F_4$ an explicit construction of DT codes, based on quadratic residues in a prime field, performs equally well. We show that DT codes are asymptotically good over $\F_q$. Specifically, we construct DT codes arbitrarily close to the asymptotic varshamov-Gilbert bound for codes of rate one half.
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