Best Known (44, 82, s)-Nets in Base 32
(44, 82, 240)-Net over F32 — Constructive and digital
Digital (44, 82, 240)-net over F32, using
- 6 times m-reduction [i] based on digital (44, 88, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 33, 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, 55, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 33, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(44, 82, 513)-Net in Base 32 — Constructive
(44, 82, 513)-net in base 32, using
- 14 times m-reduction [i] based on (44, 96, 513)-net in base 32, using
- base change [i] based on digital (28, 80, 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, 80, 513)-net over F64, using
(44, 82, 1105)-Net over F32 — Digital
Digital (44, 82, 1105)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3282, 1105, F32, 38) (dual of [1105, 1023, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3282, 1107, F32, 38) (dual of [1107, 1025, 39]-code), using
- 70 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 35 times 0) [i] based on linear OA(3272, 1027, F32, 38) (dual of [1027, 955, 39]-code), using
- construction XX applied to C1 = C([1022,35]), C2 = C([0,36]), C3 = C1 + C2 = C([0,35]), and C∩ = C1 ∩ C2 = C([1022,36]) [i] based on
- linear OA(3270, 1023, F32, 37) (dual of [1023, 953, 38]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,35}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3270, 1023, F32, 37) (dual of [1023, 953, 38]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,36], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3272, 1023, F32, 38) (dual of [1023, 951, 39]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,36}, and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(3268, 1023, F32, 36) (dual of [1023, 955, 37]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,35], and designed minimum distance d ≥ |I|+1 = 37 [i]
- 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
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,35]), C2 = C([0,36]), C3 = C1 + C2 = C([0,35]), and C∩ = C1 ∩ C2 = C([1022,36]) [i] based on
- 70 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 35 times 0) [i] based on linear OA(3272, 1027, F32, 38) (dual of [1027, 955, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3282, 1107, F32, 38) (dual of [1107, 1025, 39]-code), using
(44, 82, 801233)-Net in Base 32 — Upper bound on s
There is no (44, 82, 801234)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 2644 262659 469358 444290 279916 533432 231998 376786 237612 588731 684487 347401 662724 421299 378275 017344 392749 656101 139579 550472 039200 > 3282 [i]