Best Known (49−22, 49, s)-Nets in Base 32
(49−22, 49, 202)-Net over F32 — Constructive and digital
Digital (27, 49, 202)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 18, 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, 31, 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, 18, 98)-net over F32, using
(49−22, 49, 301)-Net in Base 32 — Constructive
(27, 49, 301)-net in base 32, using
- 1 times m-reduction [i] based on (27, 50, 301)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- (15, 38, 257)-net in base 32, using
- 2 times m-reduction [i] based on (15, 40, 257)-net in base 32, using
- base change [i] based on digital (0, 25, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 25, 257)-net over F256, using
- 2 times m-reduction [i] based on (15, 40, 257)-net in base 32, using
- digital (1, 12, 44)-net over F32, using
- (u, u+v)-construction [i] based on
(49−22, 49, 1079)-Net over F32 — Digital
Digital (27, 49, 1079)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3249, 1079, F32, 22) (dual of [1079, 1030, 23]-code), using
- 46 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 10 times 0, 1, 31 times 0) [i] based on linear OA(3243, 1027, F32, 22) (dual of [1027, 984, 23]-code), using
- construction XX applied to C1 = C([1022,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([1022,20]) [i] based on
- linear OA(3241, 1023, F32, 21) (dual of [1023, 982, 22]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,19}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3241, 1023, F32, 21) (dual of [1023, 982, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3243, 1023, F32, 22) (dual of [1023, 980, 23]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,20}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3239, 1023, F32, 20) (dual of [1023, 984, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [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([1022,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([1022,20]) [i] based on
- 46 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 10 times 0, 1, 31 times 0) [i] based on linear OA(3243, 1027, F32, 22) (dual of [1027, 984, 23]-code), using
(49−22, 49, 802432)-Net in Base 32 — Upper bound on s
There is no (27, 49, 802433)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 56 539424 227777 040738 079583 825065 420088 831101 253916 532704 379541 658615 092352 > 3249 [i]