Best Known (33, 68, s)-Nets in Base 32
(33, 68, 202)-Net over F32 — Constructive and digital
Digital (33, 68, 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, 44, 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
(33, 68, 288)-Net in Base 32 — Constructive
(33, 68, 288)-net in base 32, using
- 16 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, 68, 509)-Net over F32 — Digital
Digital (33, 68, 509)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3268, 509, F32, 2, 35) (dual of [(509, 2), 950, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3268, 515, F32, 2, 35) (dual of [(515, 2), 962, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3268, 1030, F32, 35) (dual of [1030, 962, 36]-code), using
- construction XX applied to C1 = C([0,33]), C2 = C([3,34]), C3 = C1 + C2 = C([3,33]), and C∩ = C1 ∩ C2 = C([0,34]) [i] based on
- linear OA(3264, 1023, F32, 34) (dual of [1023, 959, 35]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,33], and designed minimum distance d ≥ |I|+1 = 35 [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 = {3,4,…,34}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3266, 1023, F32, 35) (dual of [1023, 957, 36]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,34], and designed minimum distance d ≥ |I|+1 = 36 [i]
- 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 = {3,4,…,33}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(322, 5, F32, 2) (dual of [5, 3, 3]-code or 5-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([0,33]), C2 = C([3,34]), C3 = C1 + C2 = C([3,33]), and C∩ = C1 ∩ C2 = C([0,34]) [i] based on
- OOA 2-folding [i] based on linear OA(3268, 1030, F32, 35) (dual of [1030, 962, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(3268, 515, F32, 2, 35) (dual of [(515, 2), 962, 36]-NRT-code), using
(33, 68, 197982)-Net in Base 32 — Upper bound on s
There is no (33, 68, 197983)-net in base 32, because
- 1 times m-reduction [i] would yield (33, 67, 197983)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 69996 022430 683835 270226 094168 398638 571159 420502 700797 742703 332833 272353 659398 667124 527233 113585 650750 > 3267 [i]