Best Known (81−12, 81, s)-Nets in Base 5
(81−12, 81, 65119)-Net over F5 — Constructive and digital
Digital (69, 81, 65119)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 14)-net over F5, using
- digital (61, 73, 65105)-net over F5, using
- net defined by OOA [i] based on linear OOA(573, 65105, F5, 12, 12) (dual of [(65105, 12), 781187, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(573, 390630, F5, 12) (dual of [390630, 390557, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(573, 390633, F5, 12) (dual of [390633, 390560, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(573, 390625, F5, 12) (dual of [390625, 390552, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(565, 390625, F5, 11) (dual of [390625, 390560, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 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 X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(573, 390633, F5, 12) (dual of [390633, 390560, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(573, 390630, F5, 12) (dual of [390630, 390557, 13]-code), using
- net defined by OOA [i] based on linear OOA(573, 65105, F5, 12, 12) (dual of [(65105, 12), 781187, 13]-NRT-code), using
(81−12, 81, 390666)-Net over F5 — Digital
Digital (69, 81, 390666)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(581, 390666, F5, 12) (dual of [390666, 390585, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(580, 390664, F5, 12) (dual of [390664, 390584, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- linear OA(573, 390625, F5, 12) (dual of [390625, 390552, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(541, 390625, F5, 7) (dual of [390625, 390584, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(57, 39, F5, 4) (dual of [39, 32, 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(11) ⊂ Ce(6) [i] based on
- linear OA(580, 390665, F5, 11) (dual of [390665, 390585, 12]-code), using Gilbert–Varšamov bound and bm = 580 > Vbs−1(k−1) = 23 923436 171683 981717 012703 349083 367342 082398 909491 584481 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 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
- linear OA(580, 390664, F5, 12) (dual of [390664, 390584, 13]-code), using
- construction X with Varšamov bound [i] based on
(81−12, 81, large)-Net in Base 5 — Upper bound on s
There is no (69, 81, large)-net in base 5, because
- 10 times m-reduction [i] would yield (69, 71, large)-net in base 5, but