Best Known (54−22, 54, s)-Nets in Base 32
(54−22, 54, 240)-Net over F32 — Constructive and digital
Digital (32, 54, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (10, 21, 205)-net over F32, using
- net defined by OOA [i] based on linear OOA(3221, 205, F32, 11, 11) (dual of [(205, 11), 2234, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3221, 1026, F32, 11) (dual of [1026, 1005, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(3221, 1024, F32, 11) (dual of [1024, 1003, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(3219, 1024, F32, 10) (dual of [1024, 1005, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(3221, 1026, F32, 11) (dual of [1026, 1005, 12]-code), using
- net defined by OOA [i] based on linear OOA(3221, 205, F32, 11, 11) (dual of [(205, 11), 2234, 12]-NRT-code), using
- digital (11, 33, 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 (10, 21, 205)-net over F32, using
(54−22, 54, 514)-Net in Base 32 — Constructive
(32, 54, 514)-net in base 32, using
- base change [i] based on (23, 45, 514)-net in base 64, using
- 1 times m-reduction [i] based on (23, 46, 514)-net in base 64, using
- (u, u+v)-construction [i] based on
- (4, 15, 257)-net in base 64, using
- 1 times m-reduction [i] based on (4, 16, 257)-net in base 64, using
- base change [i] based on digital (0, 12, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 12, 257)-net over F256, using
- 1 times m-reduction [i] based on (4, 16, 257)-net in base 64, using
- (8, 31, 257)-net in base 64, using
- 1 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
- base change [i] based on digital (0, 24, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- base change [i] based on digital (0, 24, 257)-net over F256, using
- 1 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
- (4, 15, 257)-net in base 64, using
- (u, u+v)-construction [i] based on
- 1 times m-reduction [i] based on (23, 46, 514)-net in base 64, using
(54−22, 54, 2088)-Net over F32 — Digital
Digital (32, 54, 2088)-net over F32, using
(54−22, 54, 3877661)-Net in Base 32 — Upper bound on s
There is no (32, 54, 3877662)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 1897 138200 730593 801989 894940 117592 327853 303534 976588 402316 091945 171426 477591 366294 > 3254 [i]