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1. Bounding the Distance of Closest Approach to Unsafe Sets with Occupation Measures

   https://doi.org/10.1109/CDC51059.2022.9992817  

   Miller, Jared ; Sznaier, Mario ( December 2022 , 60th IEEE Conf. Decision and Control)

   This paper presents a method to lower-bound the distance of closest approach between points on an unsafe set and points along system trajectories. Such a minimal distance is a quantifiable and interpretable certificate of safety of trajectories, as compared to prior art in barrier and density methods which offers a binary indication of safety/unsafety. The distance estimation problem is converted into a infinitedimensional linear program in occupation measures based on existing work in peak estimation and optimal transport. The moment-SOS hierarchy is used to obtain a sequence of lower bounds obtained through solving semidefinite programs in increasing size, and these lower bounds will converge to the true minimal distance as the degree approaches infinity under mild conditions (e.g. Lipschitz dynamics, compact sets). 

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2. Input-output distance properties of good linear codes

   Roozbehani, Hajir ; Polyanskiy, Yury ( June 2018 , 2018 IEEE Int. Symp. Inf. Theory (ISIT))

   —Consider a linear code defined as a mapping between vector spaces of dimensions k and n. Let β∗ denote the minimal (relative) weight among all images of input vectors of full Hamming weight k. Operationally, β∗ characterizes the threshold for adversarial (erasure) noise beyond which decoder is guaranteed to produce estimate of k-input with 100% symbol error rate (SER). This paper studies the relation between β∗ and δ, the minimum distance of the code, which gives the threshold for 0% SER. An optimal tradeoff between β∗ and δ is obtained (over large alphabets) and all linear codes achieving β∗ = 1 are classified: they are repetition-like. More generally, a design criteria is proposed for codes with favorable graceful degradation properties. As an example, it is shown that in an overdetermined system of n homogeneous linear equations in k variables (over a field) it is always possible to satisfy some k − 1 equations with non-zero assignments to every unknown, provided that any subset of k equations is linearly independent. This statement is true if and only if n ≥ 2k − 1. 

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3. ELF Codes: Concatenated Codes with an Expurgating Linear Function as the Outer Code

   https://doi.org/10.1109/ISTC57237.2023.10273535  

   Wesel, R. ; Antonini, A. ; Wang, L. ; Sui, W. ; Towell, B. ; Grissett, H. ( September 2023 , IEEE)

   An expurgating linear function (ELF) is an outer code that disallows low-weight codewords of the inner code. ELFs can be designed either to maximize the minimum distance or to minimize the codeword error rate (CER) of the expurgated code. A list-decoding sieve can efficiently identify ELFs that maximize the minimum distance of the expurgated code. For convolutional inner codes, this paper provides analytical distance spectrum union (DSU) bounds on the CER of the concatenated code. For short codeword lengths, ELFs transform a good inner code into a great concatenated code. For a constant message size of K = 64 bits or constant codeword blocklength of N = 152 bits, an ELF can reduce the gap at CER 10−6 between the DSU and the random-coding union (RCU) bounds from over 1 dB for the inner code alone to 0.23 dB for the concatenated code. The DSU bounds can also characterize puncturing that mitigates the rate overhead of the ELF while maintaining the DSU-to-RCU gap. List Viterbi decoding guided by the ELF achieves maximum likelihood (ML) decoding of the concatenated code with a sufficiently large list size. The rate-K/(K+m) ELF outer code reduces rate and list decoding increases decoder complexity. As SNR increases, the average list size converges to 1 and average complexity is similar to Viterbi decoding on the trellis of the inner code. For rare large-magnitude noise events, which occur less often than the FER of the inner code, a deep search in the list finds the ML codeword. 

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4. Distance Bounds for Generalized Bicycle Codes

   https://doi.org/10.3390/sym14071348  

   Wang, Renyu ; Pryadko, Leonid P. ( July 2022 , Symmetry)

   Generalized bicycle (GB) codes is a class of quantum error-correcting codes constructed from a pair of binary circulant matrices. Unlike for other simple quantum code ansätze, unrestricted GB codes may have linear distance scaling. In addition, low-density parity-check GB codes have a naturally overcomplete set of low-weight stabilizer generators, which is expected to improve their performance in the presence of syndrome measurement errors. For such GB codes with a given maximum generator weight w, we constructed upper distance bounds by mapping them to codes local in D≤w−1 dimensions, and lower existence bounds which give d≥O(n1/2). We have also conducted an exhaustive enumeration of GB codes for certain prime circulant sizes in a family of two-qubit encoding codes with row weights 4, 6, and 8; the observed distance scaling is consistent with A(w)n1/2+B(w), where n is the code length and A(w) is increasing with w. 

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5. Finite-Blocklength and Error-Exponent Analyses for LDPC Codes in Point-to-Point and Multiple Access Communication

   https://doi.org/10.1109/ISIT44484.2020.9173963  

   Liu, Yuxin ; Effros, Michelle ( June 2020 , 2020 IEEE International Symposium on Information Theory (ISIT))

   null (Ed.)

   This paper applies error-exponent and dispersionstyle analyses to derive finite-blocklength achievability bounds for low-density parity-check (LDPC) codes over the point-to-point channel (PPC) and multiple access channel (MAC). The error-exponent analysis applies Gallager's error exponent to bound achievable symmetrical and asymmetrical rates in the MAC. The dispersion-style analysis begins with a generalization of the random coding union (RCU) bound from random code ensembles with i.i.d. codewords to random code ensembles in which codewords may be statistically dependent; this generalization is useful since the codewords of random linear codes such as LDPC codes are dependent. Application of the RCU bound yields finite-blocklength error bounds and asymptotic achievability results for both i.i.d. random codes and LDPC codes. For discrete, memoryless channels, these results show that LDPC codes achieve first- and second-order performance that is optimal for the PPC and identical to the best prior results for the MAC. 

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