Best Known (66, 66+30, s)-Nets in Base 5
(66, 66+30, 262)-Net over F5 — Constructive and digital
Digital (66, 96, 262)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 16, 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 (50, 80, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 40, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 40, 126)-net over F25, using
- digital (1, 16, 10)-net over F5, using
(66, 66+30, 639)-Net over F5 — Digital
Digital (66, 96, 639)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(596, 639, F5, 30) (dual of [639, 543, 31]-code), using
- 6 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0) [i] based on linear OA(595, 632, F5, 30) (dual of [632, 537, 31]-code), using
- construction XX applied to C1 = C([623,27]), C2 = C([0,28]), C3 = C1 + C2 = C([0,27]), and C∩ = C1 ∩ C2 = C([623,28]) [i] based on
- linear OA(591, 624, F5, 29) (dual of [624, 533, 30]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,27}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(591, 624, F5, 29) (dual of [624, 533, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(595, 624, F5, 30) (dual of [624, 529, 31]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,28}, and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(587, 624, F5, 28) (dual of [624, 537, 29]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(50, 4, F5, 0) (dual of [4, 4, 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(50, 4, F5, 0) (dual of [4, 4, 1]-code) (see above)
- construction XX applied to C1 = C([623,27]), C2 = C([0,28]), C3 = C1 + C2 = C([0,27]), and C∩ = C1 ∩ C2 = C([623,28]) [i] based on
- 6 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0) [i] based on linear OA(595, 632, F5, 30) (dual of [632, 537, 31]-code), using
(66, 66+30, 47754)-Net in Base 5 — Upper bound on s
There is no (66, 96, 47755)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 12 622609 808186 464545 822045 122636 448675 099781 115536 190086 654421 449525 > 596 [i]