Galois ring

The Galois ring is [a5] the unique Galois extension of of degree . For instance is and is . Generalizing finite fields (cf. Finite field), these rings find applications in similar areas: linear recurrences [a1], [a6], [a7], cyclic codes [a9], [a2] [a3], association schemes [a10], and character sums [a10], [a4]. For a connection with Witt rings see [a8] (cf. also Witt ring). Two different constructions of these rings are given below: bottom-up, starting from a finite field, and top-down, starting from a local field.

Bottom up.

This is the first and the most algorithmic one. Let . Pick an irreducible monic primitive polynomial of degree , as in the standard construction of from , and lift it to a polynomial over in such a way that the nice finite field property ( divides ) still holds. In the language of linear recurrences (or linear feedback shift registers), one has lifted an -sequence of period over into a linear recurrence over of the same period. This is construction of [a1]. Note that an arbitrary lift will lead to multiplying the period by a power of , as in construction of [a1]. For example, , gives a period and not . Now, let

Top down.

This -adic approach was introduced in print in [a4] but was implicitly known to M. Yamada [a10], who used the term "Teichmüller" , as in -adic analysis, and also to E. Spiegel [a9]. Denote by the ring of -adic integers, best viewed as the set of formal expansions in powers of with coefficients in the residue field . Then . For higher values of one considers the unramified extension of generated by (an -th root of unity) and its ring of integers . Let denote the set of roots of over this latter ring. This set of so-called Teichmüller representatives reduces modulo to . The ring of integers of the -adic field admits the following expansion: , which converges in the sense of the -adic valuation. Modulo this yields

Multiplicative structure.

The ring comprises units and zero divisors . The multiplicative group is the direct product of by the group of so-called principal units . The group of principal units is isomorphic, for or , to the additive group of . The Galois group of over is isomorphic to the Galois group of over and therefore cyclic of order .

References

[a1]	S. Boztas, A.R. Hammons, P.V. Kumar, "-phase sequence with near optimum correlation properties" IEEE Inform. Th. , 38 (1992) pp. 1101–1113
[a2]	A. Bonnecaze, P. Solé, A.R. Calderbank, "Quaternary construction of unimodular lattices" IEEE Inform. Th. , 41 (1995) pp. 366–376
[a3]	A.R. Hammons, P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, P. Solé, "The -linearity of Kerdock, Preparata, Goethals, and related codes" IEEE Trans. Information Th. , 40 (1994) pp. 301–319
[a4]	V. Kumar, T. Helleseth, R.A. Calderbank, "An upper bound for Weil-type exponential sums over Galois rings and applications" IEEE Inform. Th. , 41 (1995)
[a5]	B.R. MacDonald, "Finite rings with identity" , M. Dekker (1974)
[a6]	P. Solé, "A quaternary cyclic code and a family of quaternary sequences with low correlation" G. Cohen (ed.) J. Wolfmann (ed.) , Coding Theory and Applications , Lecture Notes in Computer Science , 388 , Springer (1989) pp. 193–201
[a7]	P. Udaya, M.U. Siddiqui, "Optimal biphase sequences with large linear complexity derived from sequences over " IEEE Inform. Th. , IT–42 (1996) pp. 202–216
[a8]	A.G. Shanbag, P.V. Kumar, T. Helleseth, "An upperbound for the extended Kloosterman sums over Galois rings" , Finite Fields and Applications (to appear)
[a9]	E. Spiegel, "Codes over revisited" Inform. and Control , 37 (1978) pp. 100–104
[a10]	M. Yamada, "Distance regular graphs of girth over an extension ring of " Graphs and Combinatorics , 6 (1980) pp. 381–384