On the Shape of the General Error Locator Polynomial for Cyclic Codes

General error locator polynomials were introduced in 2005 as an alternative decoding for cyclic codes. We now present a conjecture on their sparsity, which would imply polynomial-time decoding for all cyclic codes. A general result on the explicit form of the general error locator polynomial for all cyclic codes is given, along with several results for specific code families, providing evidence to our conjecture. From these, a theoretical justification of the sparsity of general error locator polynomials is obtained for all binary cyclic codes with <inline-formula> <tex-math notation="LaTeX">$t\leq 2$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$n<105$ </tex-math></inline-formula>, as well as for <inline-formula> <tex-math notation="LaTeX">$t=3$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$n<63$ </tex-math></inline-formula>, except for some cases where the conjectured sparsity is proved by a computer check. Moreover, we summarize all related results, previously published, and we show how they provide further evidence to our conjecture. Finally, we discuss the link between our conjecture and the complexity of bounded-distance decoding of the cyclic codes.