Best Known (117, 117+29, s)-Nets in Base 5
(117, 117+29, 1118)-Net over F5 — Constructive and digital
Digital (117, 146, 1118)-net over F5, using
- 51 times duplication [i] based on digital (116, 145, 1118)-net over F5, using
- net defined by OOA [i] based on linear OOA(5145, 1118, F5, 29, 29) (dual of [(1118, 29), 32277, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(5145, 15653, F5, 29) (dual of [15653, 15508, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(5145, 15655, F5, 29) (dual of [15655, 15510, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- linear OA(5139, 15625, F5, 29) (dual of [15625, 15486, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(5115, 15625, F5, 24) (dual of [15625, 15510, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(5145, 15655, F5, 29) (dual of [15655, 15510, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(5145, 15653, F5, 29) (dual of [15653, 15508, 30]-code), using
- net defined by OOA [i] based on linear OOA(5145, 1118, F5, 29, 29) (dual of [(1118, 29), 32277, 30]-NRT-code), using
(117, 117+29, 15472)-Net over F5 — Digital
Digital (117, 146, 15472)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5146, 15472, F5, 29) (dual of [15472, 15326, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(5146, 15639, F5, 29) (dual of [15639, 15493, 30]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(5145, 15626, F5, 31) (dual of [15626, 15481, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(5133, 15626, F5, 27) (dual of [15626, 15493, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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([0,15]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5146, 15639, F5, 29) (dual of [15639, 15493, 30]-code), using
(117, 117+29, large)-Net in Base 5 — Upper bound on s
There is no (117, 146, large)-net in base 5, because
- 27 times m-reduction [i] would yield (117, 119, large)-net in base 5, but