Best Known (107−21, 107, s)-Nets in Base 5
(107−21, 107, 1568)-Net over F5 — Constructive and digital
Digital (86, 107, 1568)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (76, 97, 1562)-net over F5, using
- net defined by OOA [i] based on linear OOA(597, 1562, F5, 21, 21) (dual of [(1562, 21), 32705, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(597, 15621, F5, 21) (dual of [15621, 15524, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(597, 15625, F5, 21) (dual of [15625, 15528, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(597, 15625, F5, 21) (dual of [15625, 15528, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(597, 15621, F5, 21) (dual of [15621, 15524, 22]-code), using
- net defined by OOA [i] based on linear OOA(597, 1562, F5, 21, 21) (dual of [(1562, 21), 32705, 22]-NRT-code), using
- digital (0, 10, 6)-net over F5, using
(107−21, 107, 15662)-Net over F5 — Digital
Digital (86, 107, 15662)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5107, 15662, F5, 21) (dual of [15662, 15555, 22]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5103, 15655, F5, 21) (dual of [15655, 15552, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(597, 15625, F5, 21) (dual of [15625, 15528, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(5103, 15658, F5, 19) (dual of [15658, 15555, 20]-code), using Gilbert–Varšamov bound and bm = 5103 > Vbs−1(k−1) = 33991 648155 809588 076181 086724 730334 880932 325183 587962 436298 912740 804325 [i]
- linear OA(51, 4, F5, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(5103, 15655, F5, 21) (dual of [15655, 15552, 22]-code), using
- construction X with Varšamov bound [i] based on
(107−21, 107, large)-Net in Base 5 — Upper bound on s
There is no (86, 107, large)-net in base 5, because
- 19 times m-reduction [i] would yield (86, 88, large)-net in base 5, but