Best Known (33, 33+32, s)-Nets in Base 32
(33, 33+32, 202)-Net over F32 — Constructive and digital
Digital (33, 65, 202)-net over F32, using
- 2 times m-reduction [i] based on digital (33, 67, 202)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 24, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (9, 43, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- digital (7, 24, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(33, 33+32, 288)-Net in Base 32 — Constructive
(33, 65, 288)-net in base 32, using
- 19 times m-reduction [i] based on (33, 84, 288)-net in base 32, using
- base change [i] based on digital (9, 60, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 60, 288)-net over F128, using
(33, 33+32, 618)-Net over F32 — Digital
Digital (33, 65, 618)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3265, 618, F32, 32) (dual of [618, 553, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3265, 1033, F32, 32) (dual of [1033, 968, 33]-code), using
- construction XX applied to C1 = C([1020,27]), C2 = C([0,28]), C3 = C1 + C2 = C([0,27]), and C∩ = C1 ∩ C2 = C([1020,28]) [i] based on
- linear OA(3261, 1023, F32, 31) (dual of [1023, 962, 32]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−3,−2,…,27}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3257, 1023, F32, 29) (dual of [1023, 966, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(3263, 1023, F32, 32) (dual of [1023, 960, 33]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−3,−2,…,28}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3255, 1023, F32, 28) (dual of [1023, 968, 29]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(322, 8, F32, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([1020,27]), C2 = C([0,28]), C3 = C1 + C2 = C([0,27]), and C∩ = C1 ∩ C2 = C([1020,28]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3265, 1033, F32, 32) (dual of [1033, 968, 33]-code), using
(33, 33+32, 285651)-Net in Base 32 — Upper bound on s
There is no (33, 65, 285652)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 68 353193 269022 673537 938169 481561 229924 457329 238767 934044 384013 973574 908086 702547 015507 067200 739069 > 3265 [i]