Best Known (110−52, 110, s)-Nets in Base 32
(110−52, 110, 240)-Net over F32 — Constructive and digital
Digital (58, 110, 240)-net over F32, using
- t-expansion [i] based on digital (51, 110, 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, 70, 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
(110−52, 110, 513)-Net in Base 32 — Constructive
(58, 110, 513)-net in base 32, using
- 322 times duplication [i] based on (56, 108, 513)-net in base 32, using
- t-expansion [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
- t-expansion [i] based on (46, 108, 513)-net in base 32, using
(110−52, 110, 1162)-Net over F32 — Digital
Digital (58, 110, 1162)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32110, 1162, F32, 52) (dual of [1162, 1052, 53]-code), using
- 125 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 18 times 0, 1, 34 times 0, 1, 54 times 0) [i] based on linear OA(32100, 1027, F32, 52) (dual of [1027, 927, 53]-code), using
- construction XX applied to C1 = C([1022,49]), C2 = C([0,50]), C3 = C1 + C2 = C([0,49]), and C∩ = C1 ∩ C2 = C([1022,50]) [i] based on
- linear OA(3298, 1023, F32, 51) (dual of [1023, 925, 52]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,49}, and designed minimum distance d ≥ |I|+1 = 52 [i]
- linear OA(3298, 1023, F32, 51) (dual of [1023, 925, 52]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,50], and designed minimum distance d ≥ |I|+1 = 52 [i]
- linear OA(32100, 1023, F32, 52) (dual of [1023, 923, 53]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,50}, and designed minimum distance d ≥ |I|+1 = 53 [i]
- linear OA(3296, 1023, F32, 50) (dual of [1023, 927, 51]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,49], and designed minimum distance d ≥ |I|+1 = 51 [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([1022,49]), C2 = C([0,50]), C3 = C1 + C2 = C([0,49]), and C∩ = C1 ∩ C2 = C([1022,50]) [i] based on
- 125 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 18 times 0, 1, 34 times 0, 1, 54 times 0) [i] based on linear OA(32100, 1027, F32, 52) (dual of [1027, 927, 53]-code), using
(110−52, 110, 794083)-Net in Base 32 — Upper bound on s
There is no (58, 110, 794084)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 3685 549653 632254 144760 571067 762551 450033 307929 301136 165292 993950 879977 995244 036402 060870 826050 282190 423880 189399 035251 193123 891272 910749 526924 138693 734741 776841 973900 > 32110 [i]