Best Known (51, 51+13, s)-Nets in Base 5
(51, 51+13, 2606)-Net over F5 — Constructive and digital
Digital (51, 64, 2606)-net over F5, using
- 52 times duplication [i] based on digital (49, 62, 2606)-net over F5, using
- net defined by OOA [i] based on linear OOA(562, 2606, F5, 13, 13) (dual of [(2606, 13), 33816, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(562, 15637, F5, 13) (dual of [15637, 15575, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(562, 15639, F5, 13) (dual of [15639, 15577, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(561, 15626, F5, 13) (dual of [15626, 15565, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(549, 15626, F5, 11) (dual of [15626, 15577, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(51, 13, F5, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(562, 15639, F5, 13) (dual of [15639, 15577, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(562, 15637, F5, 13) (dual of [15637, 15575, 14]-code), using
- net defined by OOA [i] based on linear OOA(562, 2606, F5, 13, 13) (dual of [(2606, 13), 33816, 14]-NRT-code), using
(51, 51+13, 12356)-Net over F5 — Digital
Digital (51, 64, 12356)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(564, 12356, F5, 13) (dual of [12356, 12292, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(564, 15642, F5, 13) (dual of [15642, 15578, 14]-code), using
- 2 times code embedding in larger space [i] based on linear OA(562, 15640, F5, 13) (dual of [15640, 15578, 14]-code), using
- construction X4 applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(561, 15626, F5, 13) (dual of [15626, 15565, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(549, 15626, F5, 11) (dual of [15626, 15577, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(513, 14, F5, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,5)), using
- dual of repetition code with length 14 [i]
- linear OA(51, 14, F5, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to C([0,6]) ⊂ C([0,5]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(562, 15640, F5, 13) (dual of [15640, 15578, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(564, 15642, F5, 13) (dual of [15642, 15578, 14]-code), using
(51, 51+13, large)-Net in Base 5 — Upper bound on s
There is no (51, 64, large)-net in base 5, because
- 11 times m-reduction [i] would yield (51, 53, large)-net in base 5, but