Construction-\(A\) code[1] 


Sphere packing constructed from a binary \((n,K)\) code using Construction \(A\) [2]. Each binary codeword \(c\) of the code is mapped to an infinite set of points \(x\) such that \(x = c\) modulo two. If the underlying binary code is linear, then the resulting set of points forms a lattice.




  • Binary code — Each binary code yields a sphere packing under Construction \(A\).
  • Linear binary code — Every binary linear code yields a lattice code under Construction \(A\).
  • Best \((10,40,4)\) code — Using Construction \(A\), the Best code yields \(P_{10c}\), a non-lattice sphere packing in 10 dimensions that is the densest known [3][2; pg. 140].
  • Julin-Golay code — Using Construction \(A\), the Julin-Golay codes yield non-lattice sphere-packings that hold records in 9 and 11 dimensions.


J. Leech, “Notes on Sphere Packings”, Canadian Journal of Mathematics 19, 251 (1967) DOI
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
J. Leech and N. J. A. Sloane, “Sphere Packings and Error-Correcting Codes”, Canadian Journal of Mathematics 23, 718 (1971) DOI
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Zoo Code ID: construction_a

Cite as:
“Construction-\(A\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_construction_a, title={Construction-\(A\) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Construction-\(A\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.