Best Known (106−18, 106, s)-Nets in Base 5
(106−18, 106, 8684)-Net over F5 — Constructive and digital
Digital (88, 106, 8684)-net over F5, using
- net defined by OOA [i] based on linear OOA(5106, 8684, F5, 18, 18) (dual of [(8684, 18), 156206, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(5106, 78156, F5, 18) (dual of [78156, 78050, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5106, 78160, F5, 18) (dual of [78160, 78054, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(599, 78125, F5, 18) (dual of [78125, 78026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(571, 78125, F5, 13) (dual of [78125, 78054, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(57, 35, F5, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 44, F5, 4) (dual of [44, 37, 5]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(5106, 78160, F5, 18) (dual of [78160, 78054, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(5106, 78156, F5, 18) (dual of [78156, 78050, 19]-code), using
(106−18, 106, 65674)-Net over F5 — Digital
Digital (88, 106, 65674)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5106, 65674, F5, 18) (dual of [65674, 65568, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5106, 78155, F5, 18) (dual of [78155, 78049, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(599, 78125, F5, 18) (dual of [78125, 78026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(578, 78125, F5, 14) (dual of [78125, 78047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(571, 78125, F5, 13) (dual of [78125, 78054, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(55, 28, F5, 3) (dual of [28, 23, 4]-code or 28-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- linear OA(50, 2, F5, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to Ce(17) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(5106, 78155, F5, 18) (dual of [78155, 78049, 19]-code), using
(106−18, 106, large)-Net in Base 5 — Upper bound on s
There is no (88, 106, large)-net in base 5, because
- 16 times m-reduction [i] would yield (88, 90, large)-net in base 5, but