Best Known (89, 107, s)-Nets in Base 5
(89, 107, 8684)-Net over F5 — Constructive and digital
Digital (89, 107, 8684)-net over F5, using
- 51 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(5106, 8684, F5, 18, 18) (dual of [(8684, 18), 156206, 19]-NRT-code), using
(89, 107, 72626)-Net over F5 — Digital
Digital (89, 107, 72626)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5107, 72626, F5, 18) (dual of [72626, 72519, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5107, 78148, F5, 18) (dual of [78148, 78041, 19]-code), using
- construction X4 applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(5106, 78125, F5, 19) (dual of [78125, 78019, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(522, 23, F5, 22) (dual of [23, 1, 23]-code or 23-arc in PG(21,5)), using
- dual of repetition code with length 23 [i]
- linear OA(51, 23, F5, 1) (dual of [23, 22, 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 X4 applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(5107, 78148, F5, 18) (dual of [78148, 78041, 19]-code), using
(89, 107, large)-Net in Base 5 — Upper bound on s
There is no (89, 107, large)-net in base 5, because
- 16 times m-reduction [i] would yield (89, 91, large)-net in base 5, but