Tensor-product code

Tensor-product code[1–4]

Also known as Tensor code, Kroneckerian code, Product code.

Description

A matrix-based code constructed out of two linear binary or \(q\)-ary codes \(C_A,C_B\) in an outer-product construction denoted as \(C_A \otimes C_B\). Its dual is sometimes called the check-product code, denoted as \(C_{A}\boxplus C_{B}\).

Codewords are those matrices whose column vectors are in \(C_A=[n_A,k_A,d_A]\) and whose row vectors are in \(C_B=[n_B,k_B,d_B]\). Codewords \(c\) of a tensor code satisfy the parity check equation \(H_A c H^{\text{T}}_B = 0\).

A check-product code forms the matrix subspace dual to its corresponding tensor-product code, \begin{align} \label{eq:dual-tensor} C_{A}\boxplus C_{B}=C_A^\perp \otimes GF(2)^{n_B} + GF(2)^{n_A} \otimes C_B^\perp~. \tag*{(1)}\end{align} The parity-check matrix of this code is \(H_A \otimes H_B\), where \(H_{A,B}\) is the parity-check matrix of \(C_{A,B}\) [5; Lemma 3.3].

Protection

For linear codes \(C_A=[n_A,k_A,d_A]\) and \(C_B=[n_B,k_B,d_B]\), the resulting tensor code is \(C_A \otimes C_B=[n_A n_B,k_A k_B,d_A d_B]\). Tensor codes can be useful for protecting against burst errors [6,7].

Many (but not all [8]) tensor codes are robustly testable [9–11], a property useful for constructing LTCs [12], including a family of \(c^3\)-LTCs [13]. A property equivalent to robust testability is \(\kappa\)-product expansion [14]. For check-product codes, this property means that for every codeword \(c_1 + c_2 \in C_{A}\boxplus C_{B}\), split up according to (1), \begin{align} \kappa\left(\frac{\|c_{1}\|_{A}}{n_{A}}+\frac{\|c_{2}\|_{B}}{n_{B}}\right)\leq\frac{|c_{1}+c_{2}|}{n_{A}n_{B}}~, \tag*{(2)}\end{align} where \(\|c_{1}\|_{A}\) (\(\|c_{2}\|_{B}\)) is the number of non-zero columns (rows) in \(c_1\) (\(c_2\)).

Check-product codes formed by two random linear codes are robustly testable [15; Thm. 1], a property useful for constructing asymptotically good QLDPC codes [15,16] and proving distance bounds [17].

Rate

Rate of the tensor-product code \(C_A \otimes C_B\) is a product of the rates of the codes \(C_A\) and \(C_B\).

Decoding

The simple decoding algorithm (first decode all columns with \(C_1\), then all rows with \(C_2\)) corrects up to \((d_A d_B-1)/4 \) errors.Algorithms such as generalized minimum-distance decoding [18] or the min-sum algorithm can decode all errors of weight up to \((d_A d_B-1)/2\). Error location may be coupled with Viterbi decoding for every faulty sub-block [19].

Realizations

Construction can be used in magnetic recording by taking the tensor product of an RS code and a parity-check code [19].

Notes

See Refs. ([20], Ch. 18; [21]) for expositions.

Parents

Matrix-based code
Parallel concatenated code — Tensor-product codes are examples of parallel concatenation [22].
Concatenated code — Tensor-product codes can be viewed as serial concatenated codes [22].

Cousins

Left-right Cayley complex code — Left-right Cayley complex codewords for a fixed graph vertex are codewords of a tensor code.
Low-density parity-check (LDPC) code — Tensor products of random LDPC codes are robustly testable [10,11].
Reed-Solomon (RS) code — Tensor codes constructed from RS codes are robustly testable [23].
Classical-product code — Tensor-product codes are utilized in classical-product code constructions.
Dinur-Hsieh-Lin-Vidick (DHLV) code — Tensor codes are used in constructing quantum DHLV codes.
Quantum check-product code — Quantum check-product codes extend the concept of a check product, which yields the dual of a tensor code, to a product between a classical and a quantum code.
Quantum Tanner code — Tensor codes are used in constructing quantum Tanner codes.
Quantum tensor-product code

References

[1]: P. Elias, “Error-free Coding”, Transactions of the IRE Professional Group on Information Theory 4, 29 (1954) DOI
[2]: H. Burton and E. Weldon, “Cyclic product codes”, IEEE Transactions on Information Theory 11, 433 (1965) DOI
[3]: G. D. Forney, Jr (1966). Concatenated Codes. MIT Press, Cambridge, MA.
[4]: W. Gore, “Further results on product codes”, IEEE Transactions on Information Theory 16, 446 (1970) DOI
[5]: A. Cross, Z. He, A. Natarajan, M. Szegedy, and G. Zhu, “Quantum Locally Testable Code with Constant Soundness”, Quantum 8, 1501 (2024) arXiv:2209.11405 DOI
[6]: L. Bahl and R. Chien, “Single- and multiple-burst-correcting properties of a class of cyclic product codes”, IEEE Transactions on Information Theory 17, 594 (1971) DOI
[7]: R. Chien and S. Ng, “Dual product codes for correction of multiple low-density burst errors”, IEEE Transactions on Information Theory 19, 672 (1973) DOI
[8]: P. Valiant, “The Tensor Product of Two Codes Is Not Necessarily Robustly Testable”, Lecture Notes in Computer Science 472 (2005) DOI
[9]: E. Ben-Sasson and M. Sudan, “Robust Locally Testable Codes and Products of Codes”, (2004) arXiv:cs/0408066
[10]: I. Dinur, M. Sudan, and A. Wigderson, “Robust Local Testability of Tensor Products of LDPC Codes”, Lecture Notes in Computer Science 304 (2006) DOI
[11]: E. Ben-Sasson and M. Viderman, “Tensor Products of Weakly Smooth Codes Are Robust”, Lecture Notes in Computer Science 290 (2008) DOI
[12]: I. Dinur, “The PCP theorem by gap amplification”, Journal of the ACM 54, 12 (2007) DOI
[13]: I. Dinur, S. Evra, R. Livne, A. Lubotzky, and S. Mozes, “Locally Testable Codes with constant rate, distance, and locality”, (2021) arXiv:2111.04808
[14]: P. Panteleev and G. Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”, (2022) arXiv:2111.03654
[15]: G. Kalachev and P. Panteleev, “Two-sided Robustly Testable Codes”, (2023) arXiv:2206.09973
[16]: I. Dinur, M.-H. Hsieh, T.-C. Lin, and T. Vidick, “Good Quantum LDPC Codes with Linear Time Decoders”, (2022) arXiv:2206.07750
[17]: A. Leverrier and G. Zémor, “Decoding quantum Tanner codes”, (2022) arXiv:2208.05537
[18]: G. Forney, “Generalized minimum distance decoding”, IEEE Transactions on Information Theory 12, 125 (1966) DOI
[19]: P. Chaichanavong and P. H. Siegel, “Tensor-product parity code for magnetic recording”, IEEE Transactions on Magnetics 42, 350 (2006) DOI
[20]: F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[21]: Wolf, Jack Keil. "An introduction to tensor product codes and applications to digital storage systems." 2006 IEEE Information Theory Workshop-ITW 2006 Chengdu. IEEE, 2006.
[22]: A. Barg and G. Zemor, “Concatenated Codes: Serial and Parallel”, IEEE Transactions on Information Theory 51, 1625 (2005) DOI
[23]: A. Polishchuk and D. A. Spielman, “Nearly-linear size holographic proofs”, Proceedings of the twenty-sixth annual ACM symposium on Theory of computing - STOC ’94 194 (1994) DOI

Page edit log

Victor V. Albert (2022-08-09) — most recent
Shashank Sule (2022-04-21)

Cite as:

“Tensor-product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/tensor

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/matrices/tensor.yml.