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].
Parent
Children
- Sloane-Whitehead code
- Vasilyev code
- Reed-Muller (RM) code — All RM codes can be constructed via the \((u|u+v)\) construction [4; Ch. 13].
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].
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.
Page edit log
- Victor V. Albert (2023-03-31) — most recent
Cite as:
“\((u|u+v)\)-construction code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/uplusv
Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/uplusv.yml.