Best Known (92−28, 92, s)-Nets in Base 5
(92−28, 92, 264)-Net over F5 — Constructive and digital
Digital (64, 92, 264)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 16, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- digital (48, 76, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 38, 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, 38, 126)-net over F25, using
- digital (2, 16, 12)-net over F5, using
(92−28, 92, 681)-Net over F5 — Digital
Digital (64, 92, 681)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(592, 681, F5, 28) (dual of [681, 589, 29]-code), using
- 46 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 12 times 0, 1, 22 times 0) [i] based on linear OA(587, 630, F5, 28) (dual of [630, 543, 29]-code), using
- construction XX applied to C1 = C([623,25]), C2 = C([0,26]), C3 = C1 + C2 = C([0,25]), and C∩ = C1 ∩ C2 = C([623,26]) [i] based on
- linear OA(585, 624, F5, 27) (dual of [624, 539, 28]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,25}, and designed minimum distance d ≥ |I|+1 = 28 [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(587, 624, F5, 28) (dual of [624, 537, 29]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,26}, and designed minimum distance d ≥ |I|+1 = 29 [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(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, 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 (see above)
- construction XX applied to C1 = C([623,25]), C2 = C([0,26]), C3 = C1 + C2 = C([0,25]), and C∩ = C1 ∩ C2 = C([623,26]) [i] based on
- 46 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 12 times 0, 1, 22 times 0) [i] based on linear OA(587, 630, F5, 28) (dual of [630, 543, 29]-code), using
(92−28, 92, 59231)-Net in Base 5 — Upper bound on s
There is no (64, 92, 59232)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 20195 166052 857283 407155 983373 029336 526808 816567 679062 197786 996225 > 592 [i]