On efficient high-order spectral-null codes over them-ary alphabet
Raffaele Mascella, Luca G. Tallini · Journal of Discrete Mathematical Sciences and Cryptography · 2005
Let be the set of all words of length N over the m-polar alphabet Φ m ={−(m−1),−(m−3),…,+(m−3),+(m−1)}, having a q-th order spectral-null at zero frequency. Any subset of is a spectral-null code of length N and order q. This paper gives an equivalent formulation of in terms of codes over the m-ary alphabet ℤ m ={0, 1,…, m−1}, derives a recursive expression for the cardinality of , shows combinatorial properties of , gives new simple ways to obtain systematic m-ary q-th order spectral-null codes, and finally, presents new efficient recursive design methods to encode k m-ary information digits into second-order spectral-null codes over ℤ m of length N(k)=n(k)+N(⌈log m n(k)(n(k)−1)/2⌉+1), whose implementing algorithm requires T=O(mk log m k) m-ary digit operations and S=O(k) m-ary digit storing elements.