initial condition is an element of finite field gf 2 m there are 2 mr possible choices for a set of initial conditions in a generic use of the modulation scheme according to embodiments of the invention a set of constants u03b1 u03b1 0 u03b1 1 u03b1 2 u03b1 r is predefined the corresponding decoder also uses this same set of constants in operation a set of initial conditions is found for each codeword being generated in order to make that codeword best match a set of required properties the choice of the initial conditions does not affect the operation of the decoder which is one of the interesting features of this type of modulation scheme the error propagation length of this modulation scheme is rm 1 bits since one corrupted bit will affect r m tuples in a preferred embodiment the modulation scheme is used in two parts in a first part an intermediate codeword a is calculated with the above transform from b with initial conditions set to zero thus in a second part of the modulation scheme a map m is defined that relates specific initial condition sets to specific criteria satisfied by the intermediate codeword a once the set of initial conditions is selected rather than recalculating the final codeword c by applying the above transform with the selected initial condition set the effect t t 1 r t 2 r t 0 t 1 t 2 t p of the initial condition set is simply added to the intermediate codeword a i e c a t this is possible because.