communication system developers

Abstract: Binary Goppa codes are important in coding theory and cryptography. It is well-known that the minimum distance of a binary Goppa code defined by a squarefree polynomial of degree $r$ is at least $2r+1$. Though from the classical results of C. J. Moreno, O. Moreno, F. Levy-dit-Vehel and S. Litsyn that the true minimum distances of binary Goppa codes are $2r+1$, when the extension field degree $m$ is larger than $(r-1)^{4r}$. There have only very few families of binary Goppa codes with determined true minimum distances. In this paper, we give necessary and sufficient conditions for binary Goppa codes such that their minimum distances are $2r+1$ and $2r+2$. Infinitely many infinite families of binary Goppa codes with minimum distance $2r+1$ are given. In particular, we construct such codes for every Goppa degree $r$ in the range $2^{h-1}<r<2^h$ over $\mathbb F_{2^{2h}}$. For every $2\le r\le\lfloor2^m/3\rfloor$, we also obtain at least $2^m-3r+1$ distinct monic squarefree Goppa polynomials of degree $r$ over $\mathbb F_{2^m}$, such that their related binary Goppa codes have minimum distance $2r+1$. We extend a criterion for $q$-ary Goppa codes to split auxiliary polynomials and obtain some infinite families of $q$-ary Goppa codes attaining their designed distances $r+1$. We also construct infinite families of binary Goppa codes with minimum distance exactly $2r+2$.

Abstract: Accurate and low latency channel estimation is critical for modern MIMO systems, particularly under mobility, where channels exhibit structured sparsity and strong temporal correlation. This paper proposes a time-series conditioned diffusion framework for channel estimation that performs denoising in the angular domain. Starting from least squares (LS) observations, we train a diffusion denoiser whose conditioning information is encoded by a long short-term memory (LSTM) network over a short observation sequence, enabling the model to exploit temporal dynamics beyond per-snapshot estimation. To robustly balance observation fidelity and learned generative priors across a wide signal-to-noise ratio (SNR) range, we introduce a learnable SNR-gated late-fusion shortcut that injects the network input into the final decoding stage through a sigmoid gate with trainable center and scale. To reduce inference latency, we adopt deterministic denoising diffusion implicit model (DDIM) style reverse updates with SNR-adaptive truncation and step allocation, which significantly reduces the number of reverse diffusion steps at high SNR while maintaining strong performance in low SNR regimes. Simulations on time-evolving standardized channel models demonstrate that the proposed method achieves consistent performance gains over existing diffusion-based channel estimation baselines, while retaining low latency through SNR-adaptive inference.