On the sharpness of a theorem of B Segre
Combinatorica
Primitive elements and polynomials with arbitrary trace
Discrete Mathematics
Finite fields
Low-density parity-check codes based on finite geometries: a rediscovery and new results
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
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In this paper, we study the code $$\mathbf{C}$$ which has as parity check matrix $$\mathbf{H}$$ the incidence matrix of the design of the Hermitian curve and its (q + 1)-secants. This code is known to have good performance with an iterative decoding algorithm, as shown by Johnson and Weller in (Proceedings at the ICEE Globe com conference, Sanfrancisco, CA, 2003). We shall prove that $$\mathbf{C}$$ has a double cyclic structure and that by shortening in a suitable way $$\mathbf{H}$$ it is possible to obtain new codes which have higher code-rate. We shall also present a simple way to constructing the matrix $$\mathbf{H}$$ via a geometric approach.