Best Known (83, 83+14, s)-Nets in Base 5
(83, 83+14, 55814)-Net over F5 — Constructive and digital
Digital (83, 97, 55814)-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 (75, 89, 55804)-net over F5, using
- net defined by OOA [i] based on linear OOA(589, 55804, F5, 14, 14) (dual of [(55804, 14), 781167, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(589, 390628, F5, 14) (dual of [390628, 390539, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(589, 390633, F5, 14) (dual of [390633, 390544, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(589, 390625, F5, 14) (dual of [390625, 390536, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(581, 390625, F5, 13) (dual of [390625, 390544, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(589, 390633, F5, 14) (dual of [390633, 390544, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(589, 390628, F5, 14) (dual of [390628, 390539, 15]-code), using
- net defined by OOA [i] based on linear OOA(589, 55804, F5, 14, 14) (dual of [(55804, 14), 781167, 15]-NRT-code), using
- digital (1, 8, 10)-net over F5, using
(83, 83+14, 390666)-Net over F5 — Digital
Digital (83, 97, 390666)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(597, 390666, F5, 14) (dual of [390666, 390569, 15]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(596, 390664, F5, 14) (dual of [390664, 390568, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(8) [i] based on
- linear OA(589, 390625, F5, 14) (dual of [390625, 390536, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(557, 390625, F5, 9) (dual of [390625, 390568, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(13) ⊂ Ce(8) [i] based on
- linear OA(596, 390665, F5, 13) (dual of [390665, 390569, 14]-code), using Gilbert–Varšamov bound and bm = 596 > Vbs−1(k−1) = 442541 093899 267155 526422 645710 953067 461902 050779 476243 001137 176033 [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(596, 390664, F5, 14) (dual of [390664, 390568, 15]-code), using
- construction X with Varšamov bound [i] based on
(83, 83+14, large)-Net in Base 5 — Upper bound on s
There is no (83, 97, large)-net in base 5, because
- 12 times m-reduction [i] would yield (83, 85, large)-net in base 5, but