Shaping and Finite-Blocklength Analysis of Quaternionic Alamouti-Type Codes

Abstract

We study finite $2\times2$ Alamouti-type space--time codes associated with classical maximal orders in definite quaternion algebras over $\mathbb Q$. For the family indexed by primes $q\equiv3\pmod4$, we prove that the Eisenstein case $q=3$ uniquely minimises the finite quotient-alphabet energy at fixed minimum lattice distance and alphabet size. This ordering holds for every integer $p\geq2$, including when the representatives and their common translation are jointly optimised, with an explicit positive gap for every $q>3$. For uniform finite inputs, we derive a controlled high-SNR expansion of the information variance and determine its limiting ratio to the mutual information deficit. The mutual information expansion specialises arbitrary input fading results and gives an explicit remainder. Standard random-coding and second-order achievability tools relate these quantities to reliability over independently faded Alamouti blocks. All alphabets preserve symbol-wise maximum-likelihood detection at the Alamouti level. Numerical energy, mutual information, and error-rate comparisons support the analysis. Centred comparisons with classical Alamouti, Golden, and Silver codes illustrate the trade-off between reliability, symbol rate, and decoding effort.