We present a uniform framework for constructing 3-designs from GL2(Fq)-invariant subspaces of Fq[X, Y ]k, the space of homogeneous polynomials of degree k, and the associated projective Reed–Solomon subcodes. Given such a subspace W, we associate a PGL2(Fq)-invariant family of k-subsets of P1(Fq). Whenever this family is nonempty, it forms a 3-(q+1, k, λ) design. Via the Cayley transform, the construction is reformulated on the unit circle Uq+1 ⊆ F× q2 , where the block conditions become explicit linear relations among elementary symmetric polynomials. This reformulation unifies several previously disparate constructions and simplifies a number of delicate ad hoc computations. When k ≤ q, the evaluation map on P1(Fq) identifies W with a GL2(Fq)-invariant subcode CW of the projective Reed–Solomon code. We prove that the associated block family is nonempty if and only if d(CW) = q+1−k. Under this condition, the supports of minimum-weight codewords in CW, as well as the supports of suitable fixed-weight codewords in the dual code C⊥W , yield further 3-designs. Applying this framework to the Lucas subspaces, which form a distinguished family of invariant subspaces, we obtain explicit block descriptions, classify the cases in which the defining conditions reduce to a single equation, and establish several emptiness and nonemptiness results. In particular, for q = pe and k = pm + 1, we show that the associated block family is nonempty if and only if m | e, in which case it yields the Steiner system S(3, pm +1, q+1). Finally, in the ternary case p = 3 and k = 7, we use the weight distribution of the ternary Melas code to determine the design parameters left undetermined by Xu et al.
We revisit the problem of statistical sequence matching initiated by Unnikrishnan (TIT 2015) and consider continuous observed sequences that are generated from unknown probability density functions. The task of sequence matching is to identify all matched pairs of sequences that are generated from the same distribution from two given databases, each of which contains multiple sequences. To deal with continuous observed sequences, we adopt the kernel method with the maximum mean discrepancy metric. We first consider the case where the number of matched pairs of sequences, a.k.a., the number of matches, is known. Our main contribution lies in proposing non-parametric fixed-length and sequential tests, proving that our tests are exponentially consistent and showing that our sequential test achieves a larger mismatch exponent than our fixed-length test. Subsequently, we generalize our results to the case with unknown number of matches. In this case, the number of matches can be zero and the task is to simultaneously estimate the number of matches and identify all matched pairs of sequences. Inspired by recent studies of the discrete case by Zhou et al. (TIT 2024, arXiv 2025), we propose non-parametric fixed-length and sequential tests, clarify the asymptotic intuition for the validity of both tests, bound the exponential decay rates of error probabilities, verify the superior performance of sequential tests and demonstrate the penalty of not knowing the number of matches. When specialized to the case where either database contains a single sequence, our results yield large deviations results for statistical classification of continuous sequences, generalizing previous studies of Gutman (TIT 1989) and Zhou, Tan, and Motani (Inf. Inference, 2020) for fixed-length tests and previous studies of Haghifam, Khisti and Tan (TIT 2021), and Li and Wang (TIT 2024) for sequential tests to continuous observed sequences.
We study matrix codes under poset metrics and introduce two row-wise refinements: the P-weight list and P-shape distributions, extending the classical NRT distributions to general poset metrics. We prove that these distributions admit MacWilliams identities for all linear matrix codes if and only if P is hierarchical. For hierarchical posets, we derive explicit MacWilliams and Pless identities. For symmetric hierarchical posets, we introduce a block-reversing map ψ. For ψ-self-dual matrix codes, the P-shape enumerator is invariant under the group generated by the normalized MacWilliams transformation. We compute the invariant ring explicitly and provide a general construction of ψ-self-dual matrix codes.
Composite DNA is a recent novel method to increase the information capacity of DNA-based data storage above the theoretical limit of 2 bits/symbol. In this method, every composite symbol does not store a single DNA nucleotide but a mixture of the four nucleotides in a predetermined ratio. By using different mixtures and ratios, the alphabet can be extended to have far more than four symbols compared to the naive approach. While this method enables higher data content per synthesis cycle, potentially reducing the DNA synthesis cost, it also imposes significant challenges for accurate DNA sequencing since the base-level errors can easily change the mixture of bases and their ratio, resulting in changes to the composite symbols. With this motivation, we propose efficient constrained coding techniques to enforce the biological constraints, including the runlength-limited constraint and the GC-content constraint, into every DNA synthesized oligo, regardless of the mixture of bases in each composite letter and their corresponding ratio, and regardless of the positions of the composite letters in each data sequence. This approach plays a crucial role in reducing the occurrence of errors. Our goals include computing the capacity of the constrained channel, constructing efficient encoders/decoders, and providing the best options for the composite letters to obtain capacity-approaching codes, particularly focusing on the parameters where the channel capacity is strictly larger than 2 bits/symbol. To the best of our knowledge, no such codebooks are known prior to this work. For certain codes’ parameters, our methods incur only one redundant symbol.
We study the evolution of Tsallis entropy along the heat flow and establish concavity results in arbitrary dimensions. Extending earlier one-dimensional results, we prove that Tsallis entropy is concave along the heat flow for q ∈ (0, 3] in dimension one and for q ∈ [1, 3] in higher dimensions. The upper endpoint q = 3 is sharp in every dimension. The proof is based on a nonlinear transformation of the heat equation, a sharp dimension-free functional inequality with constant Cu = 3, and a rigorous justification of the integration-by-parts identities used in the argument. The sharp inequality is proved by an explicit integration-by-parts sum-of-squares identity, rather than by a computer-assisted semidefinite-programming search. As consequences, we recover a generalized de Bruijn identity, prove monotonicity of the associated q-Fisher information along the heat flow, and establish concavity results for an exponential transform of Tsallis entropy. These results include the classical Shannon entropy-power concavity theorem at the endpoint q = 1, μ = 2. We also obtain an asymptotic concavity result for this exponential Tsallis-entropy functional for general initial data, as well as a sharp auxiliary functional inequality which may be of independent analytic interest.
