Best Known (57, 81, s)-Nets in Base 32
(57, 81, 2734)-Net over F32 — Constructive and digital
Digital (57, 81, 2734)-net over F32, using
- 321 times duplication [i] based on digital (56, 80, 2734)-net over F32, using
- net defined by OOA [i] based on linear OOA(3280, 2734, F32, 24, 24) (dual of [(2734, 24), 65536, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3280, 32808, F32, 24) (dual of [32808, 32728, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(13) [i] based on
- linear OA(3270, 32768, F32, 24) (dual of [32768, 32698, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(3240, 32768, F32, 14) (dual of [32768, 32728, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(3210, 40, F32, 9) (dual of [40, 30, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, 43, F32, 9) (dual of [43, 33, 10]-code), using
- algebraic-geometric code AG(F,33P) [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- discarding factors / shortening the dual code based on linear OA(3210, 43, F32, 9) (dual of [43, 33, 10]-code), using
- construction X applied to Ce(23) ⊂ Ce(13) [i] based on
- OA 12-folding and stacking [i] based on linear OA(3280, 32808, F32, 24) (dual of [32808, 32728, 25]-code), using
- net defined by OOA [i] based on linear OOA(3280, 2734, F32, 24, 24) (dual of [(2734, 24), 65536, 25]-NRT-code), using
(57, 81, 5462)-Net in Base 32 — Constructive
(57, 81, 5462)-net in base 32, using
- 321 times duplication [i] based on (56, 80, 5462)-net in base 32, using
- base change [i] based on digital (26, 50, 5462)-net over F256, using
- 2561 times duplication [i] based on digital (25, 49, 5462)-net over F256, using
- net defined by OOA [i] based on linear OOA(25649, 5462, F256, 24, 24) (dual of [(5462, 24), 131039, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(25649, 65544, F256, 24) (dual of [65544, 65495, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(25647, 65536, F256, 24) (dual of [65536, 65489, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(25641, 65536, F256, 21) (dual of [65536, 65495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- OA 12-folding and stacking [i] based on linear OA(25649, 65544, F256, 24) (dual of [65544, 65495, 25]-code), using
- net defined by OOA [i] based on linear OOA(25649, 5462, F256, 24, 24) (dual of [(5462, 24), 131039, 25]-NRT-code), using
- 2561 times duplication [i] based on digital (25, 49, 5462)-net over F256, using
- base change [i] based on digital (26, 50, 5462)-net over F256, using
(57, 81, 60803)-Net over F32 — Digital
Digital (57, 81, 60803)-net over F32, using
(57, 81, large)-Net in Base 32 — Upper bound on s
There is no (57, 81, large)-net in base 32, because
- 22 times m-reduction [i] would yield (57, 59, large)-net in base 32, but