 codewords that are amenable to Euclidean distance Shannon limit for a binary-input AWGN channel and decoding instead of Hamming distance decoding. a bit error rate of 10-6. Informally, a lattice A is an infinite regular array of points that covers an m-dimensional space uniformly. A 3.4.3.1.4 IRA codes lattice is defined by a set of basis vectors SO that any point Irregular repeat-accumulate (IRA) codes were in the lattice can be represented as a linear combination introduced by Jin and colleagues [27] as a generalization of the basis vectors. More precisely, if the basis vectors of the repeat-accumulate (RA) codes of [16]. IRA codes are given as V1 = V2 = (U2,1, U2,2 can be encoded in linear time. They are decoded with the v2,m), ..., Vn = Vn,m) where mzn, then we sum-product algorithm, achieving on the binary-input can define a generator matrix, G, to be a matrix that AWGN channel a performance competitive with that of contains the basis vectors as the rows of the matrix and the best LDPC codes of comparable complexity. any lattice point can be written as 3.4.3.1.5 Tornado codes A=ZG, Tornado codes [31,32] are (n, k) erasure codes that allow where of is an n-dimensional vector of integers [13]. For encoding and decoding with time complexity linear in example, a generator matrix for the m-dimensional integer the block length N. This speed-up over Reed-Solomon lattice (often written as Zm) is the m-dimensional identity codes is obtained at the cost that slightly more than k matrix. The lattice points of the m-dimensional integer encoding symbols are required to reconstruct all k lattice consist of all the possible m-dimensional vectors of information symbols. More precisely, Luby et al. [32] integers. prove that for any E > 0, one can construct a Tornado The conventional method of using a lattice for channel code that recovers all k information symbols from coding is to take a finite subset of lattice points and define only E)k encoding symbols with probability a one-to-one mapping between the lattice points and