Best Known (115, 115+31, s)-Nets in Base 5
(115, 115+31, 1042)-Net over F5 — Constructive and digital
Digital (115, 146, 1042)-net over F5, using
- net defined by OOA [i] based on linear OOA(5146, 1042, F5, 31, 31) (dual of [(1042, 31), 32156, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(5146, 15631, F5, 31) (dual of [15631, 15485, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(5146, 15632, F5, 31) (dual of [15632, 15486, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(5145, 15625, F5, 31) (dual of [15625, 15480, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(51, 7, F5, 1) (dual of [7, 6, 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 Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(5146, 15632, F5, 31) (dual of [15632, 15486, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(5146, 15631, F5, 31) (dual of [15631, 15485, 32]-code), using
(115, 115+31, 9098)-Net over F5 — Digital
Digital (115, 146, 9098)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5146, 9098, F5, 31) (dual of [9098, 8952, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(5146, 15632, F5, 31) (dual of [15632, 15486, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(5145, 15625, F5, 31) (dual of [15625, 15480, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(51, 7, F5, 1) (dual of [7, 6, 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 Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(5146, 15632, F5, 31) (dual of [15632, 15486, 32]-code), using
(115, 115+31, large)-Net in Base 5 — Upper bound on s
There is no (115, 146, large)-net in base 5, because
- 29 times m-reduction [i] would yield (115, 117, large)-net in base 5, but