Best Known (53, 101, s)-Nets in Base 32
(53, 101, 240)-Net over F32 — Constructive and digital
Digital (53, 101, 240)-net over F32, using
- t-expansion [i] based on digital (51, 101, 240)-net over F32, using
- 8 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
- 8 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
(53, 101, 513)-Net in Base 32 — Constructive
(53, 101, 513)-net in base 32, using
- t-expansion [i] based on (46, 101, 513)-net in base 32, using
- 7 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
- 7 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(53, 101, 1065)-Net over F32 — Digital
Digital (53, 101, 1065)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32101, 1065, F32, 48) (dual of [1065, 964, 49]-code), using
- discarding factors / shortening the dual code based on linear OA(32101, 1088, F32, 48) (dual of [1088, 987, 49]-code), using
- 52 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 0, 1, 6 times 0, 1, 13 times 0, 1, 26 times 0) [i] based on linear OA(3292, 1027, F32, 48) (dual of [1027, 935, 49]-code), using
- construction XX applied to C1 = C([1022,45]), C2 = C([0,46]), C3 = C1 + C2 = C([0,45]), and C∩ = C1 ∩ C2 = C([1022,46]) [i] based on
- 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(3290, 1023, F32, 47) (dual of [1023, 933, 48]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,46], and designed minimum distance d ≥ |I|+1 = 48 [i]
- linear OA(3292, 1023, F32, 48) (dual of [1023, 931, 49]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,46}, and designed minimum distance d ≥ |I|+1 = 49 [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(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([1022,45]), C2 = C([0,46]), C3 = C1 + C2 = C([0,45]), and C∩ = C1 ∩ C2 = C([1022,46]) [i] based on
- 52 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 0, 1, 6 times 0, 1, 13 times 0, 1, 26 times 0) [i] based on linear OA(3292, 1027, F32, 48) (dual of [1027, 935, 49]-code), using
- discarding factors / shortening the dual code based on linear OA(32101, 1088, F32, 48) (dual of [1088, 987, 49]-code), using
(53, 101, 682599)-Net in Base 32 — Upper bound on s
There is no (53, 101, 682600)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 104 749803 458779 252875 068147 818790 977305 799710 633832 974616 496312 418433 758358 975793 187588 738809 795835 617674 383224 727408 531789 294169 838477 917917 947039 935756 > 32101 [i]