Best Known (74, 74+15, s)-Nets in Base 5
(74, 74+15, 11164)-Net over F5 — Constructive and digital
Digital (74, 89, 11164)-net over F5, using
- 51 times duplication [i] based on digital (73, 88, 11164)-net over F5, using
- net defined by OOA [i] based on linear OOA(588, 11164, F5, 15, 15) (dual of [(11164, 15), 167372, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(588, 78149, F5, 15) (dual of [78149, 78061, 16]-code), using
- 1 times truncation [i] based on linear OA(589, 78150, F5, 16) (dual of [78150, 78061, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- 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(564, 78125, F5, 12) (dual of [78125, 78061, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(54, 25, F5, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,5)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- 1 times truncation [i] based on linear OA(589, 78150, F5, 16) (dual of [78150, 78061, 17]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(588, 78149, F5, 15) (dual of [78149, 78061, 16]-code), using
- net defined by OOA [i] based on linear OOA(588, 11164, F5, 15, 15) (dual of [(11164, 15), 167372, 16]-NRT-code), using
(74, 74+15, 76346)-Net over F5 — Digital
Digital (74, 89, 76346)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(589, 76346, F5, 15) (dual of [76346, 76257, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(589, 78152, F5, 15) (dual of [78152, 78063, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(585, 78126, F5, 15) (dual of [78126, 78041, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(557, 78126, F5, 11) (dual of [78126, 78069, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(589, 78152, F5, 15) (dual of [78152, 78063, 16]-code), using
(74, 74+15, large)-Net in Base 5 — Upper bound on s
There is no (74, 89, large)-net in base 5, because
- 13 times m-reduction [i] would yield (74, 76, large)-net in base 5, but