Transformers have achieved remarkable success across natural language processing (NLP) and computer vision (CV). However, deep transformer models often suffer from an over-smoothing issue, in which token representations converge to similar values as they pass through successive transformer blocks. In this paper, we establish an equivalence between the hidden-state dynamics induced by stacked attention layers and graph neural diffusion on a complete graph. From this perspective, over-smoothing can be interpreted as a consequence of the dissipative nature of the underlying diffusion dynamics. Motivated by this physical interpretation, we propose Wavy Transformer, which consists of a novel attention layer based on second-order wavy dynamics. We also introduce a feed-forward network and a normalization layer designed to preserve the physical state-velocity relationship under the chain rule, thereby extending the transformer architecture. We further validate our proposed techniques on various transformer models for NLP and CV tasks. The results consistently demonstrate that Wavy Transformer improves performance with minimal additional parameters and no extra hyperparameter tuning.

BCH codes are among the best practical cyclic codes widely used in consumer electronics, communication systems, and storage devices. However, not much is known about BCH codes with large minimum distance. In this paper, we consider narrow-sense BCH codes of length $n = \frac{q^m-1}{N}$ with designed distance $\delta = \frac{s}{q-1}n$ proportional to $n$, where $N$ divides $\frac{q^m-1}{q-1}$ and $1 \le s \le q-1$. We determine both their dimensions and minimum distances. In particular, when $N=1$, the codes are primitive, with minimum distance $d=\frac{s}{q-1}(q^m-1)$ and dimension $k = (q-s)^m$. The general result on code dimensions is achieved by applying generating functions and inverse discrete Fourier transforms to an enumeration problem.