By G. David Forney Jr. (auth.), Marc Fossorier, Hideki Imai, Shu Lin, Alain Poli (eds.)

This publication constitutes the refereed lawsuits of the nineteenth overseas Symposium on utilized Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-13, held in Honolulu, Hawaii, united states in November 1999.

The forty two revised complete papers provided including six invited survey papers have been conscientiously reviewed and chosen from a complete of 86 submissions. The papers are equipped in sections on codes and iterative interpreting, mathematics, graphs and matrices, block codes, earrings and fields, interpreting tools, code building, algebraic curves, cryptography, codes and deciphering, convolutional codes, designs, deciphering of block codes, modulation and codes, Gröbner bases and AG codes, and polynomials.

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**Additional info for Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 13th International Symposium, AAECC-13 Honolulu, Hawaii, USA, November 15–19, 1999 Proceedings**

**Example text**

McEliece We have therefore shown that the only possible accumulation point of the set {δ(q)} is 1/2, which proves that the limit of the δ(q)’s exists and equals 1/2. ✷ We can now prove the main theorem: Proof of Theorem 21. We have, for every q, γq = qc0 (q) = q 1 − δ(q) 1 − e−2rq (δ(q)) . 9), we obtain γq ≤ q 1 − δ(q) rq (δ(q)). δ(q) Now from Corollary 32, we know that rq (δ) ≤ (log 2)/q, so that we have γq ≤ 1 − δ(q) log 2. δ(q) But by Proposition 31, limq→∞ δ(q) = 1/2, so lim sup γq ≤ log 2.

Since any e-monomial matrix M ∈ CN ×N can be written in the form M = πdiag(ω a1 , . . , ω an ) with a permutation π ∈ SN and non-zero coeﬃcients ω a1 , . . , ω an , just the 2N integers π(1), . . , π(N ) and a1 , . . , aN have to be stored for M. For the group h G and any r ∈ IN deﬁne dr (G) := k=1 drk , where h denotes the number of conjugacy classes of G and d1 , . . , dh the degrees of the irreducible characters of G. , all matrices Di,k (gl ), 1 ≤ i ≤ n, 1 ≤ k ≤ hi (hi denoting the n number of conjugacy classes of Gi ), 1 ≤ l ≤ i, is proportional to i=1 i · d1 (Gi ), n which is bounded from above by i=1 log(|Gi |) · d1 (Gi ).

Eχn−1 M) . . )). As χi |χi+1 = 1 (by Cliﬀord’s Theorem), e(w)M is a simple CG0 -module, hence one-dimensional. (3). By Cliﬀord’s Theorem, G acts transitively on the irreducible constituents of χ ↓ Gn−1 . Observing that Gn−1 acts trivially on Irr(Gn−1 ), an induction on n yields our claim. (4). This follows from geχi g −1 = egχi , for all χi ∈ Irr(Gi ). (5). By (3) and (4), G acts transitively on the set of lines {e(w)M|w ∈ W (χ)} according to ge(w)M = e(gw)gM = e(gw)M. Choosing any nonzero vector xw ∈ e(w)M yields a basis (xw )w∈W (χ) of M, and, by (2), the corresponding matrix representation is monomial.