Best Known (84−39, 84, s)-Nets in Base 32
(84−39, 84, 240)-Net over F32 — Constructive and digital
Digital (45, 84, 240)-net over F32, using
- 7 times m-reduction [i] based on digital (45, 91, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 34, 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, 57, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 34, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(84−39, 84, 513)-Net in Base 32 — Constructive
(45, 84, 513)-net in base 32, using
- 18 times m-reduction [i] based on (45, 102, 513)-net in base 32, using
- base change [i] based on digital (28, 85, 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, 85, 513)-net over F64, using
(84−39, 84, 1107)-Net over F32 — Digital
Digital (45, 84, 1107)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3284, 1107, F32, 39) (dual of [1107, 1023, 40]-code), using
- 70 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 35 times 0) [i] based on linear OA(3274, 1027, F32, 39) (dual of [1027, 953, 40]-code), using
- construction XX applied to C1 = C([1022,36]), C2 = C([0,37]), C3 = C1 + C2 = C([0,36]), and C∩ = C1 ∩ C2 = C([1022,37]) [i] based on
- linear OA(3272, 1023, F32, 38) (dual of [1023, 951, 39]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,36}, and designed minimum distance d ≥ |I|+1 = 39 [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(3274, 1023, F32, 39) (dual of [1023, 949, 40]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,37}, and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3270, 1023, F32, 37) (dual of [1023, 953, 38]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,36], and designed minimum distance d ≥ |I|+1 = 38 [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([1022,36]), C2 = C([0,37]), C3 = C1 + C2 = C([0,36]), and C∩ = C1 ∩ C2 = C([1022,37]) [i] based on
- 70 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 35 times 0) [i] based on linear OA(3274, 1027, F32, 39) (dual of [1027, 953, 40]-code), using
(84−39, 84, 961564)-Net in Base 32 — Upper bound on s
There is no (45, 84, 961565)-net in base 32, because
- 1 times m-reduction [i] would yield (45, 83, 961565)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 84616 361670 519743 599068 475249 409763 394292 226579 658849 438573 457098 882481 806423 615871 300727 689679 727034 732361 469282 358712 191148 > 3283 [i]