Best Known (43, 43+40, s)-Nets in Base 32
(43, 43+40, 240)-Net over F32 — Constructive and digital
Digital (43, 83, 240)-net over F32, using
- 2 times m-reduction [i] based on digital (43, 85, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 32, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 11 and N(F) ≥ 120, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- digital (11, 53, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 32, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(43, 43+40, 513)-Net in Base 32 — Constructive
(43, 83, 513)-net in base 32, using
- 7 times m-reduction [i] based on (43, 90, 513)-net in base 32, using
- base change [i] based on digital (28, 75, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- base change [i] based on digital (28, 75, 513)-net over F64, using
(43, 43+40, 840)-Net over F32 — Digital
Digital (43, 83, 840)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3283, 840, F32, 40) (dual of [840, 757, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3283, 1045, F32, 40) (dual of [1045, 962, 41]-code), using
- construction XX applied to C1 = C([1018,33]), C2 = C([3,34]), C3 = C1 + C2 = C([3,33]), and C∩ = C1 ∩ C2 = C([1018,34]) [i] based on
- linear OA(3274, 1023, F32, 39) (dual of [1023, 949, 40]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−5,−4,…,33}, and designed minimum distance d ≥ |I|+1 = 40 [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(3276, 1023, F32, 40) (dual of [1023, 947, 41]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−5,−4,…,34}, and designed minimum distance d ≥ |I|+1 = 41 [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(327, 20, F32, 7) (dual of [20, 13, 8]-code or 20-arc in PG(6,32)), using
- discarding factors / shortening the dual code based on linear OA(327, 32, F32, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
- Reed–Solomon code RS(25,32) [i]
- discarding factors / shortening the dual code based on linear OA(327, 32, F32, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), 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([1018,33]), C2 = C([3,34]), C3 = C1 + C2 = C([3,33]), and C∩ = C1 ∩ C2 = C([1018,34]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3283, 1045, F32, 40) (dual of [1045, 962, 41]-code), using
(43, 43+40, 472397)-Net in Base 32 — Upper bound on s
There is no (43, 83, 472398)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 84616 839348 128450 133930 050114 808428 265805 646902 966966 128065 069051 161266 524643 106000 756306 389579 967384 186813 392717 012973 695196 > 3283 [i]