Best Known (111, 111+29, s)-Nets in Base 5
(111, 111+29, 1116)-Net over F5 — Constructive and digital
Digital (111, 140, 1116)-net over F5, using
- 51 times duplication [i] based on digital (110, 139, 1116)-net over F5, using
- net defined by OOA [i] based on linear OOA(5139, 1116, F5, 29, 29) (dual of [(1116, 29), 32225, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [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]
- OOA 14-folding and stacking with additional row [i] based on linear OA(5139, 15625, F5, 29) (dual of [15625, 15486, 30]-code), using
- net defined by OOA [i] based on linear OOA(5139, 1116, F5, 29, 29) (dual of [(1116, 29), 32225, 30]-NRT-code), using
(111, 111+29, 10814)-Net over F5 — Digital
Digital (111, 140, 10814)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5140, 10814, F5, 29) (dual of [10814, 10674, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(5140, 15638, F5, 29) (dual of [15638, 15498, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [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(5127, 15625, F5, 27) (dual of [15625, 15498, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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 Ce(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(5140, 15638, F5, 29) (dual of [15638, 15498, 30]-code), using
(111, 111+29, large)-Net in Base 5 — Upper bound on s
There is no (111, 140, large)-net in base 5, because
- 27 times m-reduction [i] would yield (111, 113, large)-net in base 5, but