Best Known (116, 116+29, s)-Nets in Base 5
(116, 116+29, 1118)-Net over F5 — Constructive and digital
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
(116, 116+29, 14575)-Net over F5 — Digital
Digital (116, 145, 14575)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5145, 14575, F5, 29) (dual of [14575, 14430, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(5145, 15651, F5, 29) (dual of [15651, 15506, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(25) ⊂ 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(5121, 15625, F5, 26) (dual of [15625, 15504, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(53, 23, F5, 2) (dual of [23, 20, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- linear OA(51, 3, F5, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- Reed–Solomon code RS(4,5) [i]
- discarding factors / shortening the dual code based on linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- construction XX applied to Ce(28) ⊂ Ce(25) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(5145, 15651, F5, 29) (dual of [15651, 15506, 30]-code), using
(116, 116+29, large)-Net in Base 5 — Upper bound on s
There is no (116, 145, large)-net in base 5, because
- 27 times m-reduction [i] would yield (116, 118, large)-net in base 5, but