Best Known (46, 46+39, s)-Nets in Base 32
(46, 46+39, 240)-Net over F32 — Constructive and digital
Digital (46, 85, 240)-net over F32, using
- 9 times m-reduction [i] based on digital (46, 94, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 35, 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, 59, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 35, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(46, 46+39, 513)-Net in Base 32 — Constructive
(46, 85, 513)-net in base 32, using
- 23 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
- base change [i] based on digital (28, 90, 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, 90, 513)-net over F64, using
(46, 46+39, 1169)-Net over F32 — Digital
Digital (46, 85, 1169)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3285, 1169, F32, 39) (dual of [1169, 1084, 40]-code), using
- 131 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 35 times 0, 1, 60 times 0) [i] based on linear OA(3274, 1027, F32, 39) (dual of [1027, 953, 40]-code), using
- construction XX applied to C1 = C([1022,36]), C2 = C([0,37]), C3 = C1 + C2 = C([0,36]), and C∩ = C1 ∩ C2 = C([1022,37]) [i] based on
- 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(3272, 1023, F32, 38) (dual of [1023, 951, 39]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,37], and designed minimum distance d ≥ |I|+1 = 39 [i]
- 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 = {−1,0,…,37}, and designed minimum distance d ≥ |I|+1 = 40 [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(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([1022,36]), C2 = C([0,37]), C3 = C1 + C2 = C([0,36]), and C∩ = C1 ∩ C2 = C([1022,37]) [i] based on
- 131 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 35 times 0, 1, 60 times 0) [i] based on linear OA(3274, 1027, F32, 39) (dual of [1027, 953, 40]-code), using
(46, 46+39, 1153977)-Net in Base 32 — Upper bound on s
There is no (46, 85, 1153978)-net in base 32, because
- 1 times m-reduction [i] would yield (46, 84, 1153978)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 2 707692 584388 702050 212244 926838 548999 365769 949834 565839 189520 708284 645278 540179 239535 157961 174409 107552 469624 563593 118585 970812 > 3284 [i]