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Topics in algebraic coding theory. (English) Zbl 0708.94013

Geometries, codes and cryptography, CISM Courses Lect. 313, 77-99 (1990).
[For the entire collection see Zbl 0699.00030.]
A wide range of topics in algebraic coding theory is surveyed. Starting with the definition of a code, Heise discusses bounds on the size of a code, cyclic, BCH, Goppa, Hamming, MDS, and RM codes, McEliece’s cryptosystem based on Goppa codes, the connection between MDS codes and geometries and the automorphism group of a code. All this is done in a mere fifteen pages: very fast for the non-expert, while the expert will find little news.
The last chapter is entirely devoted to QR-codes. It includes the Golay codes, the Gleason-Prange theorem (on the automorphism group of a QR code) and the square root bound (on the minimum distance of a QR code). A recent improvement on the square root bound is stated and proved. It applies to QR codes of length \(n=3\) (mod 4) over a prime field GF(p) with \(p=3\) (mod 4). This improvement might be unknown to quite a few people.
Reviewer: L.M.G.M.Tolhuizen

MSC:

94B05 Linear codes (general theory)
94B65 Bounds on codes

Citations:

Zbl 0699.00030