GRAND features both soft-input and hard-input variants that are well suited to efficient hardware implementations that can be characterized with achievable average and worst-case decoding latency. This paper introduces step-GRAND, a soft-input variant of GRAND that, in addition to achieving appealing average decoding latency, also reduces the worst-case decoding latency of the corresponding hardware implementation. The hardware implementation results demonstrate that the proposed step-GRAND can decode CA-polar code (128,105+11) with an average information throughput of 47.7 Gbps at the target FER of
<10−7. Furthermore, the proposed step-GRAND hardware is 10 × more area efficient than the previous soft-input ORBGRAND hardware implementation, and its worst-case latency is
681×that of the previous ORBGRAND hardware.
A random access code (RAC) encodes an
L-bit string into a
k-bit message,
L>k, so that any requested bit can be recovered with high probability; a quantum RAC (QRAC) uses
k qubits instead. We give a geometric characterization of optimal classical
(L,k)-RACs under average and worst-case decoding criteria. The average criterion is reduced to choosing
2k representatives in
{0,1}L, while the worst-case criterion is reduced to a minimax problem over
2k points in
[0,1]L with a distance-like objective. This framework proves optimality for several parameter families, with many optimal constructions arising from standard infinite families of binary linear codes. It also yields two explicit classical--quantum separations. First, for every
L>1, we construct a
(L,1)-QRAC whose average decoding success probability strictly exceeds the optimal classical value. Second, for the family
(2k−1,k), we prove worst-case optimality of a classical RAC and construct a QRAC with strictly larger worst-case success probability. For the family
(L,L−1), the framework identifies a classical RAC that is average-case optimal and, under a stated conjecture, also worst-case optimal. The same viewpoint further recovers explicit
(L,L−1)-QRACs attaining a previously conjectured upper-bound value.
In this article we revisit smoothing bounds in parallel between lattices and codes. Initially introduced by Micciancio and Regev, these bounds were instantiated with Gaussian distributions and were crucial for arguing the security of many lattice-based cryptosystems. Unencumbered by direct application concerns, we provide a systematic study of how these bounds are obtained for both lattices and codes, transferring techniques between both areas. We also consider multiple choices of spherically symmetric noise distributions. We found that the best strategy for a worst-case bound combines Parseval’s Identity, the Cauchy-Schwarz inequality, and the second linear programming bound, and this holds for both codes and lattices and all noise distributions at hand. For an average-case analysis, the linear programming bound can be replaced by an expected value computation. This alone gives optimal results for spherically uniform noise over random codes and random lattices. This also improves prior Gaussian smoothing bounds for worst-case lattices, but surprisingly this provides even better results with uniform ball noise than for Gaussian (or Bernoulli noise for codes). This counterintuitive situation can be resolved by adequate decomposition and truncation of Gaussian and Bernoulli distributions into a superposition of uniform noise, giving further improvement for those cases, and putting them on par with the uniform cases.