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In general, the generator matrix sparsity is a critical factor in determining the encoding complexity of a linear code. Further, certain applications, e.g., distributed crowdsourcing schemes utilizing linear codes, require most or even all the columns of the generator matrix to have some degree of sparsity. In this paper, we leverage polar codes and the well-established channel polarization to design capacity-achieving codes with a certain constraint on the weights of all the columns in the generator matrix (GM) while having a low-complexity decoding algorithm. We first show that given a binary-input memoryless symmetric (BMS) channel $$W$$ and a constant $$s \in (0, 1]$$ , there exists a polarization kernel such that the corresponding polar code is capacity-achieving with the rate of polarization $s/2$ , and the GM column weights being bounded from above by $$N^{s}$$ . To improve the sparsity versus error rate trade-off, we devise a column-splitting algorithm and two coding schemes for BEC and then for general BMS channels. The polar-based codes generated by the two schemes inherit several fundamental properties of polar codes with the original $$2 \times 2$$ kernel including the decay in error probability, decoding complexity, and the capacity-achieving property. Furthermore, they demonstrate the additional property that their GM column weights are bounded from above sublinearly in $$N$$ , while the original polar codes have some column weights that are linear in $$N$$ . In particular, for any BEC and $$\beta < 0.5$$ , the existence of a sequence of capacity-achieving polar-based codes where all the GM column weights are bounded from above by $$N^{\lambda} $$ with $$\lambda \approx 0.585$$ , and with the error probability bounded by $${\mathcal {O}}(2^{-N^{\beta }})$$ under a decoder with complexity $${\mathcal {O}}(N\log N)$$ , is shown. The existence of similar capacity-achieving polar-based codes with the same decoding complexity is shown for any BMS channel and $$\beta < 0.5$$ with $$\lambda \approx 0.631$$ .
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This content will become publicly available on July 1, 2027
Efficient Multiple Error Correction in Binary HDC Systems Using Majority-Encoded Hypervector Matrices
Binary Hyperdimensional Computing primitives o‚er signi€cant energy and hardware bene€ts for power constrained edge and portable AI applications. However, these are vulnerable to timing and so… errors in the associa tive memory that stores high-dimensional data representa tions. ‡is is made worse under aggressive power conditions, such as from voltage overscaling, resulting in signi€cant performance loss. In this research, class hypervectors are represented in matrix form and the matrix columns are ordered in terms of criticality (with respect to classi€cation accuracy). Subsequently, cosine similarity is used to cluster the critical matrix columns into submatrices, one submatrix for each cluster. Traditional algorithmic checksums su‚er from aliasing due to their binary representations. To address this, we use checks based on permutations of the columns of the submatrix for error detection. For error correction, we introduce a novel majority-vote column reconstruction (MVCR) algorithm, where the erroneous submatrix columns are re placed with a column vector representing a majority vote across all the column vectors of the submatrix. In contrast, erroneous noncritical matrix columns are suppressed to zero. ‡eproposedapproach is validated on SRAM-based platforms under voltage scaling, achieving up to 4X improvement in error resilience over state-of-the-art methods with minimal overhead.
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- Award ID(s):
- 2414361
- PAR ID:
- 10686265
- Publisher / Repository:
- IEEE
- Date Published:
- Subject(s) / Keyword(s):
- Error resilience, Hyperdimensional computing, Associative memory
- Format(s):
- Medium: X
- Location:
- Italy
- Sponsoring Org:
- National Science Foundation
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