Best Known (82, 100, s)-Nets in Base 7
(82, 100, 13080)-Net over F7 — Constructive and digital
Digital (82, 100, 13080)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- digital (73, 91, 13072)-net over F7, using
- net defined by OOA [i] based on linear OOA(791, 13072, F7, 18, 18) (dual of [(13072, 18), 235205, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(791, 117648, F7, 18) (dual of [117648, 117557, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(791, 117649, F7, 18) (dual of [117649, 117558, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(791, 117649, F7, 18) (dual of [117649, 117558, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(791, 117648, F7, 18) (dual of [117648, 117557, 19]-code), using
- net defined by OOA [i] based on linear OOA(791, 13072, F7, 18, 18) (dual of [(13072, 18), 235205, 19]-NRT-code), using
- digital (0, 9, 8)-net over F7, using
(82, 100, 117690)-Net over F7 — Digital
Digital (82, 100, 117690)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7100, 117690, F7, 18) (dual of [117690, 117590, 19]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(798, 117686, F7, 18) (dual of [117686, 117588, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(791, 117649, F7, 18) (dual of [117649, 117558, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(761, 117649, F7, 12) (dual of [117649, 117588, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(77, 37, F7, 5) (dual of [37, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(798, 117688, F7, 17) (dual of [117688, 117590, 18]-code), using Gilbert–Varšamov bound and bm = 798 > Vbs−1(k−1) = 182 399347 647397 568005 117739 585778 309681 584529 696909 753113 273226 663512 594091 425911 [i]
- linear OA(70, 2, F7, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(798, 117686, F7, 18) (dual of [117686, 117588, 19]-code), using
- construction X with Varšamov bound [i] based on
(82, 100, large)-Net in Base 7 — Upper bound on s
There is no (82, 100, large)-net in base 7, because
- 16 times m-reduction [i] would yield (82, 84, large)-net in base 7, but