Best Known (90−41, 90, s)-Nets in Base 32
(90−41, 90, 240)-Net over F32 — Constructive and digital
Digital (49, 90, 240)-net over F32, using
- 13 times m-reduction [i] based on digital (49, 103, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 38, 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, 65, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 38, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(90−41, 90, 513)-Net in Base 32 — Constructive
(49, 90, 513)-net in base 32, using
- t-expansion [i] based on (46, 90, 513)-net in base 32, using
- 18 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
- 18 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(90−41, 90, 1263)-Net over F32 — Digital
Digital (49, 90, 1263)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3290, 1263, F32, 41) (dual of [1263, 1173, 42]-code), using
- 222 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 18 times 0, 1, 35 times 0, 1, 61 times 0, 1, 88 times 0) [i] based on linear OA(3279, 1030, F32, 41) (dual of [1030, 951, 42]-code), using
- construction XX applied to C1 = C([1021,37]), C2 = C([0,38]), C3 = C1 + C2 = C([0,37]), and C∩ = C1 ∩ C2 = C([1021,38]) [i] based on
- 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 = {−2,−1,…,37}, and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(3274, 1023, F32, 39) (dual of [1023, 949, 40]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,38], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3278, 1023, F32, 41) (dual of [1023, 945, 42]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,38}, and designed minimum distance d ≥ |I|+1 = 42 [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(321, 5, F32, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- Reed–Solomon code RS(31,32) [i]
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), 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([1021,37]), C2 = C([0,38]), C3 = C1 + C2 = C([0,37]), and C∩ = C1 ∩ C2 = C([1021,38]) [i] based on
- 222 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 18 times 0, 1, 35 times 0, 1, 61 times 0, 1, 88 times 0) [i] based on linear OA(3279, 1030, F32, 41) (dual of [1030, 951, 42]-code), using
(90−41, 90, 1336160)-Net in Base 32 — Upper bound on s
There is no (49, 90, 1336161)-net in base 32, because
- 1 times m-reduction [i] would yield (49, 89, 1336161)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 90 855461 106254 075227 219942 906380 929544 373722 868470 293252 318570 660955 769475 330992 800274 799508 312748 051293 049431 271124 972680 666062 261944 > 3289 [i]