Methods and apparatus for encoding data

A method of encoding data words comprising the step of encoding said data words as 2m-symbol code words where m≧3, each code word symbol having q different possible values where q is an even integer greater than 1, for transmission using multi-carrier frequency division multiplexing, such as coded orthogonal frequency division multiplexing (COFDM), wherein said data words are encoded in accordance with their value and code words selected in complete cosets, for a peak-to-mean envelope power ratio of said transmission not greater than a predetermined threshold of 2p+1 for any fixed integer p where 1≦p≦m−2, from a set of one or more cosets of a linear sub-code of a code having a generator matrix as follows:

• ( ( 2.0 + 1 ) × 2 m ) / 2 m ( ( 2.1 + 1 ) × 2 m ) / 2 m … ( ( 2.0 + 1 ) × 2 m - 1 ) / 2 m - 1 ( ( 2.1 + 1 ) × 2 m - 1 ) / 2 m - 1 … … ( ( 2.0 + 1 ) × 2 y ) / 2 y ( ( 2.1 + 1 ) × 2 y ) / 2 y … … ( ( 2.0 + 1 ) × 2 1 ) / 2 1 ( ( 2.1 + 1 ) × 2 1 ) / 2 1 … ( ( 2.0 + 1 ) × 2 0 ) / 2 0 ( ( 2.1 + 1 ) × 2 0 ) / 2 0 … ⁢ ( ( 2. ⁢ z + 1 ) × 2 m ) / 2 m … ( ( 2. ⁢ ( 2 m - 1 ) + 1 ) × 2 m ) / 2 m ( ( 2. ⁢ z + 1 ) × 2 m - 1 ) / 2 m - 1 … ( ( 2. ⁢ ( 2 m - 1 ) + 1 ) × 2 m - 1 ) / 2 m - 1 ( ( 2. ⁢ z + 1 ) × 2 y ) / 2 y … ( ( 2. ⁢ ( 2 m - 1 ) + 1 ) × 2 y ) ⁢ 2 y ( ( 2. ⁢ z + 1 ) × 2 1 ) / 2 1 … ( ( 2. ⁢ ( 2 m - 1 ) + 1 ) × 2 1 ) / 2 1 ( ( 2. ⁢ z + 1 ) × 2 0 ) / 2 0 … ( ( 2. ⁢ ( 2 m - 1 ) + 1 ) × 2 0 ) / 2 0
• where z has integer values from 0 to 2m−1, y has integer values from m to 0, x indicates bitwise multiplication of factors expressed as (m+1)-bit binary numbers, and the result of division is expressed as a single digit;
• wherein all coset representatives of cosets in said set of cosets have the form ( q / 2 ) ⁢ ⁢ ( X 1 * X 2 ) ⊕ ( q / 2 ) ⁢ ( X 2 * X 3 ) ⊕ … ⊕ ( q / 2 ) ⁢ ( X m - p - 1 * X m - p ) ⊕ ⁢ v 1 , 1 ⁡ ( X 1 * X m - p + 1 ) ⊕ v 1 , 2 ⁡ ( X 2 * X m - p + 1 ) ⊕ … ⊕ v 1 , m - p ⁡ ( X m - p * X m - p + 1 ) ⊕ ⁢ v 2 , 1 ⁡ ( X 1 * X m - p + 2 ) ⊕ v 2 , 2 ⁡ ( X 2 * X m - p + 2 ) ⊕ … ⊕ v 2 , m - p ⁡ ( X m - p * X m - p + 2 ) ⊕ ⁢ ⋯ ⁢ v p , 1 ⁡ ( X 1 * X m ) ⊕ v p , 2 ⁡ ( X 2 * X m ) ⊕ … ⊕ v p , m - p ⁡ ( X m - p * X m ) ⊕ ⁢ w 1 , 2 ⁡ ( X m - p + 1 * X m - p + 2 ) ⊕ ⁢ w 1 , 3 ⁡ ( X m - p + 1 * X m - p + 3 ) ⊕ w 2 , 3 ⁡ ( X m - p + 2 * X m - p + 3 ) ⊕ ⁢ w 1 , 4 ⁡ ( X m - p + 1 * X m - p + 4 ) ⊕ w 2 , 4 ⁡ ( X m - p + 2 * X m - p + 4 ) ⊕ ⁢ w 3 , 4 ⁡ ( X m - p + 3 * X m - p + 4 ) ⊕ ⁢ ⋯ ⁢ w 1 , p ⁡ ( X m - p + 1 * X m ) ⊕ w 2 , p ⁡ ( X m - p + 2 * X m ) ⊕ … ⊕ w p - 1 , p ⁡ ( X m - 1 * X m )
• or a form derived therefrom by a permutation of the subscripts of the terms X1 to Xm, where X1 to Xm are row labels of the first m rows in the generator matrix, terms of the form Vp,m−p(Xm−p*xm) indicate derivation of a word by symbol-wise multiplication of rows such as Xm−p and Xm followed by symbol-wise multiplication modulo q of such word by a respective coefficient such as vp,m−p, p indicates symbol-wise addition of words modulo q, and the coefficients vi,j for 1≦i≦p and 1≦j≦m−p and the coefficients wi,j for 1≦i<j≦p have integer values chosen from the range 0 to q−1.