Best Known (71, 87, s)-Nets in Base 5
(71, 87, 9766)-Net over F5 — Constructive and digital
Digital (71, 87, 9766)-net over F5, using
- 51 times duplication [i] based on digital (70, 86, 9766)-net over F5, using
- net defined by OOA [i] based on linear OOA(586, 9766, F5, 16, 16) (dual of [(9766, 16), 156170, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(586, 78128, F5, 16) (dual of [78128, 78042, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(586, 78133, F5, 16) (dual of [78133, 78047, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [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(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(51, 8, F5, 1) (dual of [8, 7, 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(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(586, 78133, F5, 16) (dual of [78133, 78047, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(586, 78128, F5, 16) (dual of [78128, 78042, 17]-code), using
- net defined by OOA [i] based on linear OOA(586, 9766, F5, 16, 16) (dual of [(9766, 16), 156170, 17]-NRT-code), using
(71, 87, 39067)-Net over F5 — Digital
Digital (71, 87, 39067)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(587, 39067, F5, 2, 16) (dual of [(39067, 2), 78047, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(587, 78134, F5, 16) (dual of [78134, 78047, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(587, 78135, F5, 16) (dual of [78135, 78048, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(13) ⊂ Ce(12) [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(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(51, 9, F5, 1) (dual of [9, 8, 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
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(15) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(587, 78135, F5, 16) (dual of [78135, 78048, 17]-code), using
- OOA 2-folding [i] based on linear OA(587, 78134, F5, 16) (dual of [78134, 78047, 17]-code), using
(71, 87, large)-Net in Base 5 — Upper bound on s
There is no (71, 87, large)-net in base 5, because
- 14 times m-reduction [i] would yield (71, 73, large)-net in base 5, but