Decoding in weighted combinatorial and other metrics

对于许多分组码，存在低复杂度硬输入有界距离解码器。然而，低复杂度软输入解码仍然是一个具有挑战性的任务。我们的目标是为有内存的信道提出一种分组码的软输入解码。对于无记忆信道，这种软输入解码被称为广义最小距离（GMD）解码，是由Forney在1966年提出的。GMD译码利用接收符号的信度，在信度加权的汉明度量中可以看作是一种有界距离译码。汉明度量匹配离散的无记忆对称信道，即，在这些信道中的最大似然解码等效于汉明度量中的最小距离解码。GMD解码以多试验方式使用分组码的硬输入纠错和擦除解码器，其中在每次解码试验中，在解码之前擦除一些最不可靠的接收符号。数以百计的出版物表明，GMD译码是一种通用程序（适用于任意码），具有非常好的性能和低复杂度。因此，它对于不同的无内存信道和解码（广义）级联码具有许多实际应用。

There exist low-complexity hard-input bounded distance decoders for many block codes. However, low complexity soft-input decoding remains still a challenging task. Our goal is to propose a soft-input decoding of block codes for channels with memory. For memoryless channels such soft-input decoding, called Generalized Minimum Distance (GMD) decoding, was suggested by Forney in 1966. GMD decoding uses reliabilities of received symbols and can be considered as a bounded distance decoding in a weighted (by the reliabilities) Hamming metric. The Hamming metric matches discrete memoryless symmetric channels, i.e., maximum-likelihood decoding in such channels is equivalent to minimum distance decoding in the Hamming metric. GMD decoding uses a hardinput error-and-erasure decoder of the block code in a multitrial manner, where in each decoding trial, a number of least reliable received symbols are erased before decoding. Hundreds of publications show that GMD decoding is a universal procedure (applicable to an arbitrary code) with very good performance and low complexity. As a result, it has many practical applications for different memoryless channels and for the decoding (generalized) concatenated codes.