Description
Code constructed using a concatenation procedure that takes in two \(q\)-ary codes \(C_1,C_2\) and produces a new code whose codewords are \((u|u+v)\) for all \(u\in C_1\) and \(v\in C_2\). If the two codes have parameters \((n,K_1,d_1)\) and \((n,K_2,d_2)\), then the output code is a \((2n,K_1 K_2, \min\{2d_1,d_2\})\) code [3; Thm. 5.10][4; pg. 76]. When \(C_1\) and \(C_2\) are linear \([n,k_1,d_1]_q\) and \([n,k_2,d_2]_q\) codes, the resulting code is linear with parameters \([2n,k_1+k_2,\min\{2d_1,d_2\}]_q\).Cousins
- \([7,4,3]\) Hamming code— Starting with the \([6,3,3]\) shortened Hamming code and applying the \((u|u+v)\) construction recursively using the repetition code yields a family of \([2^m,m+1,2^{m-1}]\) codes for \(m\geq1\) that saturate the Griesmer bound [3; pg. 90].
- Binary quadratic-residue (QR) code— The \((u|u+v)\) construction can be used to obtain nonlinear binary quadratic-residue codes [2].
- Berman code— Berman codes are recursively constructed via a construction that is similar to the \((u|u+v)\) construction [5].
Member of code lists
Primary Hierarchy
Parents
\((u|u+v)\)-construction code
Children
All RM codes can be constructed via the \((u|u+v)\) construction [4; Ch. 13].
References
- [1]
- M. Plotkin, “Binary codes with specified minimum distance”, IEEE Transactions on Information Theory 6, 445 (1960) DOI
- [2]
- N. Sloane and D. Whitehead, “New family of single-error correcting codes”, IEEE Transactions on Information Theory 16, 717 (1970) DOI
- [3]
- J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
- [4]
- F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)
- [5]
- L. P. Natarajan and P. Krishnan, “Berman Codes: A Generalization of Reed-Muller Codes that Achieve BEC Capacity”, (2023) arXiv:2202.09981
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2023-03-31)
Cite as:
“\((u|u+v)\)-construction code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/uplusv, arXiv:2606.11484