Best Known (58, 74, s)-Nets in Base 7
(58, 74, 2109)-Net over F7 — Constructive and digital
Digital (58, 74, 2109)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 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 (50, 66, 2101)-net over F7, using
- net defined by OOA [i] based on linear OOA(766, 2101, F7, 16, 16) (dual of [(2101, 16), 33550, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(766, 16808, F7, 16) (dual of [16808, 16742, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(766, 16812, F7, 16) (dual of [16812, 16746, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- linear OA(766, 16807, F7, 16) (dual of [16807, 16741, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(761, 16807, F7, 15) (dual of [16807, 16746, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(70, 5, F7, 0) (dual of [5, 5, 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
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(766, 16812, F7, 16) (dual of [16812, 16746, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(766, 16808, F7, 16) (dual of [16808, 16742, 17]-code), using
- net defined by OOA [i] based on linear OOA(766, 2101, F7, 16, 16) (dual of [(2101, 16), 33550, 17]-NRT-code), using
- digital (0, 8, 8)-net over F7, using
(58, 74, 16841)-Net over F7 — Digital
Digital (58, 74, 16841)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(774, 16841, F7, 16) (dual of [16841, 16767, 17]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(773, 16839, F7, 16) (dual of [16839, 16766, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(9) [i] based on
- linear OA(766, 16807, F7, 16) (dual of [16807, 16741, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(741, 16807, F7, 10) (dual of [16807, 16766, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(77, 32, F7, 5) (dual of [32, 25, 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(15) ⊂ Ce(9) [i] based on
- linear OA(773, 16840, F7, 15) (dual of [16840, 16767, 16]-code), using Gilbert–Varšamov bound and bm = 773 > Vbs−1(k−1) = 131780 130096 442083 399015 830170 778250 345493 879350 105487 345399 [i]
- linear OA(70, 1, F7, 0) (dual of [1, 1, 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(773, 16839, F7, 16) (dual of [16839, 16766, 17]-code), using
- construction X with Varšamov bound [i] based on
(58, 74, large)-Net in Base 7 — Upper bound on s
There is no (58, 74, large)-net in base 7, because
- 14 times m-reduction [i] would yield (58, 60, large)-net in base 7, but