Best Known (97−45, 97, s)-Nets in Base 32
(97−45, 97, 240)-Net over F32 — Constructive and digital
Digital (52, 97, 240)-net over F32, using
- t-expansion [i] based on digital (51, 97, 240)-net over F32, using
- 12 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
- 12 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
(97−45, 97, 513)-Net in Base 32 — Constructive
(52, 97, 513)-net in base 32, using
- t-expansion [i] based on (46, 97, 513)-net in base 32, using
- 11 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
- 11 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(97−45, 97, 1188)-Net over F32 — Digital
Digital (52, 97, 1188)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3297, 1188, F32, 45) (dual of [1188, 1091, 46]-code), using
- 150 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 1, 5 times 0, 1, 10 times 0, 1, 22 times 0, 1, 41 times 0, 1, 64 times 0) [i] based on linear OA(3286, 1027, F32, 45) (dual of [1027, 941, 46]-code), using
- construction XX applied to C1 = C([1022,42]), C2 = C([0,43]), C3 = C1 + C2 = C([0,42]), and C∩ = C1 ∩ C2 = C([1022,43]) [i] based on
- linear OA(3284, 1023, F32, 44) (dual of [1023, 939, 45]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,42}, and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(3284, 1023, F32, 44) (dual of [1023, 939, 45]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,43], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(3286, 1023, F32, 45) (dual of [1023, 937, 46]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,43}, and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(3282, 1023, F32, 43) (dual of [1023, 941, 44]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,42], and designed minimum distance d ≥ |I|+1 = 44 [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([1022,42]), C2 = C([0,43]), C3 = C1 + C2 = C([0,42]), and C∩ = C1 ∩ C2 = C([1022,43]) [i] based on
- 150 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 1, 5 times 0, 1, 10 times 0, 1, 22 times 0, 1, 41 times 0, 1, 64 times 0) [i] based on linear OA(3286, 1027, F32, 45) (dual of [1027, 941, 46]-code), using
(97−45, 97, 1079975)-Net in Base 32 — Upper bound on s
There is no (52, 97, 1079976)-net in base 32, because
- 1 times m-reduction [i] would yield (52, 96, 1079976)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 3 121764 510600 584526 639107 140575 981060 596028 485892 052276 090289 466720 921162 202900 885842 233510 936410 234629 010801 840974 807259 906352 722622 741209 658234 > 3296 [i]