Best Known (45, 45+16, s)-Nets in Base 7
(45, 45+16, 308)-Net over F7 — Constructive and digital
Digital (45, 61, 308)-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 (37, 53, 300)-net over F7, using
- net defined by OOA [i] based on linear OOA(753, 300, F7, 16, 16) (dual of [(300, 16), 4747, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(753, 2400, F7, 16) (dual of [2400, 2347, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(753, 2401, F7, 16) (dual of [2401, 2348, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(753, 2401, F7, 16) (dual of [2401, 2348, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(753, 2400, F7, 16) (dual of [2400, 2347, 17]-code), using
- net defined by OOA [i] based on linear OOA(753, 300, F7, 16, 16) (dual of [(300, 16), 4747, 17]-NRT-code), using
- digital (0, 8, 8)-net over F7, using
(45, 45+16, 2975)-Net over F7 — Digital
Digital (45, 61, 2975)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(761, 2975, F7, 16) (dual of [2975, 2914, 17]-code), using
- 562 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 0, 1, 11 times 0, 1, 31 times 0, 1, 76 times 0, 1, 161 times 0, 1, 273 times 0) [i] based on linear OA(753, 2405, F7, 16) (dual of [2405, 2352, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- linear OA(753, 2401, F7, 16) (dual of [2401, 2348, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(749, 2401, F7, 15) (dual of [2401, 2352, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 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
- 562 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 0, 1, 11 times 0, 1, 31 times 0, 1, 76 times 0, 1, 161 times 0, 1, 273 times 0) [i] based on linear OA(753, 2405, F7, 16) (dual of [2405, 2352, 17]-code), using
(45, 45+16, 1743452)-Net in Base 7 — Upper bound on s
There is no (45, 61, 1743453)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 3556 158133 879640 584408 437285 716463 149081 363521 674241 > 761 [i]