Best Known (99−47, 99, s)-Nets in Base 32
(99−47, 99, 240)-Net over F32 — Constructive and digital
Digital (52, 99, 240)-net over F32, using
- t-expansion [i] based on digital (51, 99, 240)-net over F32, using
- 10 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 40, 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, 69, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 40, 120)-net over F32, using
- (u, u+v)-construction [i] based on
- 10 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
(99−47, 99, 513)-Net in Base 32 — Constructive
(52, 99, 513)-net in base 32, using
- t-expansion [i] based on (46, 99, 513)-net in base 32, using
- 9 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
- 9 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(99−47, 99, 1057)-Net over F32 — Digital
Digital (52, 99, 1057)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3299, 1057, F32, 47) (dual of [1057, 958, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(3299, 1085, F32, 47) (dual of [1085, 986, 48]-code), using
- 49 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 25 times 0) [i] based on linear OA(3290, 1027, F32, 47) (dual of [1027, 937, 48]-code), using
- construction XX applied to C1 = C([1022,44]), C2 = C([0,45]), C3 = C1 + C2 = C([0,44]), and C∩ = C1 ∩ C2 = C([1022,45]) [i] based on
- linear OA(3288, 1023, F32, 46) (dual of [1023, 935, 47]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,44}, and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(3288, 1023, F32, 46) (dual of [1023, 935, 47]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,45], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(3290, 1023, F32, 47) (dual of [1023, 933, 48]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,45}, and designed minimum distance d ≥ |I|+1 = 48 [i]
- linear OA(3286, 1023, F32, 45) (dual of [1023, 937, 46]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,44], and designed minimum distance d ≥ |I|+1 = 46 [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([1022,44]), C2 = C([0,45]), C3 = C1 + C2 = C([0,44]), and C∩ = C1 ∩ C2 = C([1022,45]) [i] based on
- 49 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 25 times 0) [i] based on linear OA(3290, 1027, F32, 47) (dual of [1027, 937, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(3299, 1085, F32, 47) (dual of [1085, 986, 48]-code), using
(99−47, 99, 787654)-Net in Base 32 — Upper bound on s
There is no (52, 99, 787655)-net in base 32, because
- 1 times m-reduction [i] would yield (52, 98, 787655)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 3196 702515 239506 262800 716454 806815 453008 375620 740131 569559 777678 350082 981654 381182 898287 220709 175701 090680 571328 832977 832897 035076 585786 371564 332096 > 3298 [i]