Celebratio Mathematica — Berlekamp

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E. R. Berlekamp, H. Rumsey, and G. Solomon: “On the solution of algebraic equations over finite fields,” Inform. Control 10 : 6 (June 1967), pp. 553–564. MR 0230706 Zbl 0166.04803 article

This article gives new fast methods for decoding certain error-correcting codes by solving certain algebraic equations. As described by Peterson [1961], the locations of a Bose–Chaudhuri–Hocquenghem code over a field of characteristic \( p \) are associated with the elements of an extension field, \( \operatorname{GF}(p^k) \). The code is designed in such a way that the weighted power-sum symmetric functions of the error locations can be obtained directly by computing appropriately chosen parity checks on the received word. Good methods for computing the elementary symmetric functions from the weighted power-sum symmetric functions have been presented by Berlekamp [1967]. The elementary symmetric functions, \( \sigma_1 \), \( \sigma_2,\dots \), \( \sigma_t = 0 \) are the coefficients of an algebraic equation whose roots are the error locations \[ xt + \sigma_1xt - 1 + \sigma_2xt - 1 + \cdots + \sigma_t = 0. \]

Previous methods for finding the roots of this equation have searched all of the elements in \( \operatorname{GF}(p^k) \) [Chien 1964] or looked up the answer in a large table [Polkinghorn 1966]. We present here improved procedures for extracting the roots of algebraic equations of small degrees.

H. Rumsey	Related
Gustave Solomon	Related