Best Known (53−27, 53, s)-Nets in Base 32
(53−27, 53, 174)-Net over F32 — Constructive and digital
Digital (26, 53, 174)-net over F32, using
- 1 times m-reduction [i] based on digital (26, 54, 174)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (5, 19, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- digital (7, 35, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (5, 19, 76)-net over F32, using
- (u, u+v)-construction [i] based on
(53−27, 53, 288)-Net in Base 32 — Constructive
(26, 53, 288)-net in base 32, using
- t-expansion [i] based on (25, 53, 288)-net in base 32, using
- 3 times m-reduction [i] based on (25, 56, 288)-net in base 32, using
- base change [i] based on digital (9, 40, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 40, 288)-net over F128, using
- 3 times m-reduction [i] based on (25, 56, 288)-net in base 32, using
(53−27, 53, 488)-Net over F32 — Digital
Digital (26, 53, 488)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3253, 488, F32, 2, 27) (dual of [(488, 2), 923, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3253, 513, F32, 2, 27) (dual of [(513, 2), 973, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3253, 1026, F32, 27) (dual of [1026, 973, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(3253, 1024, F32, 27) (dual of [1024, 971, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(3251, 1024, F32, 26) (dual of [1024, 973, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- OOA 2-folding [i] based on linear OA(3253, 1026, F32, 27) (dual of [1026, 973, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(3253, 513, F32, 2, 27) (dual of [(513, 2), 973, 28]-NRT-code), using
(53−27, 53, 191703)-Net in Base 32 — Upper bound on s
There is no (26, 53, 191704)-net in base 32, because
- 1 times m-reduction [i] would yield (26, 52, 191704)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 1 852746 601734 673005 134203 017613 225778 534620 663195 834341 988616 281443 325721 509542 > 3252 [i]