Best Known (47, 67, s)-Nets in Base 5
(47, 67, 252)-Net over F5 — Constructive and digital
Digital (47, 67, 252)-net over F5, using
- 7 times m-reduction [i] based on digital (47, 74, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 37, 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, 37, 126)-net over F25, using
(47, 67, 643)-Net over F5 — Digital
Digital (47, 67, 643)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(567, 643, F5, 20) (dual of [643, 576, 21]-code), using
- 5 step Varšamov–Edel lengthening with (ri) = (1, 4 times 0) [i] based on linear OA(566, 637, F5, 20) (dual of [637, 571, 21]-code), using
- construction XX applied to C1 = C([622,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([622,17]) [i] based on
- linear OA(561, 624, F5, 19) (dual of [624, 563, 20]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−2,−1,…,16}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(557, 624, F5, 18) (dual of [624, 567, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(565, 624, F5, 20) (dual of [624, 559, 21]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−2,−1,…,17}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(553, 624, F5, 17) (dual of [624, 571, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [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, 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
- construction XX applied to C1 = C([622,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([622,17]) [i] based on
- 5 step Varšamov–Edel lengthening with (ri) = (1, 4 times 0) [i] based on linear OA(566, 637, F5, 20) (dual of [637, 571, 21]-code), using
(47, 67, 54570)-Net in Base 5 — Upper bound on s
There is no (47, 67, 54571)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 67766 182928 869248 437595 980568 454728 838699 012441 > 567 [i]