Best Known (72, 72+14, s)-Nets in Base 5
(72, 72+14, 11171)-Net over F5 — Constructive and digital
Digital (72, 86, 11171)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (64, 78, 11161)-net over F5, using
- net defined by OOA [i] based on linear OOA(578, 11161, F5, 14, 14) (dual of [(11161, 14), 156176, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(578, 78127, F5, 14) (dual of [78127, 78049, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(578, 78132, F5, 14) (dual of [78132, 78054, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- 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(50, 7, F5, 0) (dual of [7, 7, 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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(578, 78132, F5, 14) (dual of [78132, 78054, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(578, 78127, F5, 14) (dual of [78127, 78049, 15]-code), using
- net defined by OOA [i] based on linear OOA(578, 11161, F5, 14, 14) (dual of [(11161, 14), 156176, 15]-NRT-code), using
- digital (1, 8, 10)-net over F5, using
(72, 72+14, 78162)-Net over F5 — Digital
Digital (72, 86, 78162)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(586, 78162, F5, 14) (dual of [78162, 78076, 15]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(585, 78160, F5, 14) (dual of [78160, 78075, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(8) [i] based on
- 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(550, 78125, F5, 9) (dual of [78125, 78075, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(13) ⊂ Ce(8) [i] based on
- linear OA(585, 78161, F5, 13) (dual of [78161, 78076, 14]-code), using Gilbert–Varšamov bound and bm = 585 > Vbs−1(k−1) = 1819 061854 118118 608512 126373 306680 779295 246871 936393 513665 [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(585, 78160, F5, 14) (dual of [78160, 78075, 15]-code), using
- construction X with Varšamov bound [i] based on
(72, 72+14, large)-Net in Base 5 — Upper bound on s
There is no (72, 86, large)-net in base 5, because
- 12 times m-reduction [i] would yield (72, 74, large)-net in base 5, but