Best Known (66, 66+29, s)-Nets in Base 5
(66, 66+29, 268)-Net over F5 — Constructive and digital
Digital (66, 95, 268)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (3, 17, 16)-net over F5, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 3 and N(F) ≥ 16, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- digital (49, 78, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 39, 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, 39, 126)-net over F25, using
- digital (3, 17, 16)-net over F5, using
(66, 66+29, 682)-Net over F5 — Digital
Digital (66, 95, 682)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(595, 682, F5, 29) (dual of [682, 587, 30]-code), using
- 44 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 12 times 0, 1, 26 times 0) [i] based on linear OA(592, 635, F5, 29) (dual of [635, 543, 30]-code), using
- construction XX applied to C1 = C([622,25]), C2 = C([0,26]), C3 = C1 + C2 = C([0,25]), and C∩ = C1 ∩ C2 = C([622,26]) [i] based on
- linear OA(589, 624, F5, 28) (dual of [624, 535, 29]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−2,−1,…,25}, and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(583, 624, F5, 27) (dual of [624, 541, 28]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,26], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 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 = {−2,−1,…,26}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(581, 624, F5, 26) (dual of [624, 543, 27]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 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(50, 2, F5, 0) (dual of [2, 2, 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
- construction XX applied to C1 = C([622,25]), C2 = C([0,26]), C3 = C1 + C2 = C([0,25]), and C∩ = C1 ∩ C2 = C([622,26]) [i] based on
- 44 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 12 times 0, 1, 26 times 0) [i] based on linear OA(592, 635, F5, 29) (dual of [635, 543, 30]-code), using
(66, 66+29, 74545)-Net in Base 5 — Upper bound on s
There is no (66, 95, 74546)-net in base 5, because
- 1 times m-reduction [i] would yield (66, 94, 74546)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 504879 668234 030814 376549 987672 696245 366366 064526 520494 007172 057985 > 594 [i]