By Wolfgang Ebeling

The aim of coding thought is the layout of effective platforms for the transmission of data. The mathematical remedy results in convinced finite constructions: the error-correcting codes. unusually difficulties that are fascinating for the layout of codes change into heavily with regards to difficulties studied partially prior and independently in natural arithmetic. during this booklet, examples of such connections are offered. The relation among lattices studied in quantity idea and geometry and error-correcting codes is mentioned. The ebook offers even as an advent to the idea of quintessential lattices and modular kinds and to coding theory.

within the 2d version a number of corrections were made. extra easy fabric has been incorporated to make the textual content much more self-contained. a brand new part at the automorphism crew of the Leech lattice has been additional. a few tricks to new effects were integrated. eventually, numerous new workouts were added.

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**Example text**

We may also assume that n == 4 (mod 8) because we can replace r, if necessary, by rEEl r or rEEl r EEl r EEl r. 1 then yields since r* = r and vol (JRn If) = 1. Since {)r is invariant under T, {)r ((TS)T) = -T~{)r(T). From this formula one derives But (TS)3 = 1, which is a contradiction. This proves (i). n Ir) = 1. 1. 5 r = r* and I' and finishes The Eisenstein Series We now consider some further very important examples of modular forms: Definition. Let k E Z, keven, k > 2. The series Gk(T) = L (=,n)E 1} (=,n);t(O,O) 1 (mT + n)k is called the Eisenstein series of index k.

3 + t . 'Y. 18 1 Lattices and Codes Since t . (3 > 0 and t . 'Y > 0, t . (3 < t . Q and t . 'Y < t . Q. Since t . Q was minimal, the elements (3 and 'Y can be written as linear combinations of elements of St with non-negative integer coefficients, and hence Q, too. This contradicts the assumption. 4 a· (3 s:; 0 for Q, (3 ESt, Q f- (3. Proof. Otherwise, Q . (3 = 1 and hence 'Y = Q - (3 E R. Then either 'Y E Rt or -'Y E Rt· In the first case Q = (3 + 'Y would be decomposable, in the latter case (3 = Q + (-'Y) would be decomposable, both being contradictions.

3 we need the following lemma. 18 Let r be an irreducible root lattice, and let (e1, ... , en) be a fundamental system of roots of r. Then the vectors of r* that have inner product 0 or 1 with all positive roots of r (with respect to (e1, ... , en)) form a complete set of coset representatives for r* Ir. Proof. Let y E r*, y =J 0, be a vector with y. 0: equal to 0 or 1 for all positive roots of r. If (3 denotes the highest root of r, then y . (3 = 1, because otherwise y . (3 = 0 which would imply y .