Best Known (52, 73, s)-Nets in Base 8
(52, 73, 409)-Net over F8 — Constructive and digital
Digital (52, 73, 409)-net over F8, using
- net defined by OOA [i] based on linear OOA(873, 409, F8, 21, 21) (dual of [(409, 21), 8516, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(873, 4091, F8, 21) (dual of [4091, 4018, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(873, 4096, F8, 21) (dual of [4096, 4023, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(873, 4096, F8, 21) (dual of [4096, 4023, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(873, 4091, F8, 21) (dual of [4091, 4018, 22]-code), using
(52, 73, 576)-Net in Base 8 — Constructive
(52, 73, 576)-net in base 8, using
- 1 times m-reduction [i] based on (52, 74, 576)-net in base 8, using
- trace code for nets [i] based on (15, 37, 288)-net in base 64, using
- 5 times m-reduction [i] based on (15, 42, 288)-net in base 64, using
- base change [i] based on digital (9, 36, 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, 36, 288)-net over F128, using
- 5 times m-reduction [i] based on (15, 42, 288)-net in base 64, using
- trace code for nets [i] based on (15, 37, 288)-net in base 64, using
(52, 73, 2984)-Net over F8 — Digital
Digital (52, 73, 2984)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(873, 2984, F8, 21) (dual of [2984, 2911, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(873, 4096, F8, 21) (dual of [4096, 4023, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(873, 4096, F8, 21) (dual of [4096, 4023, 22]-code), using
(52, 73, 2056482)-Net in Base 8 — Upper bound on s
There is no (52, 73, 2056483)-net in base 8, because
- 1 times m-reduction [i] would yield (52, 72, 2056483)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 105312 791546 675019 867812 357301 893969 988158 322264 098808 235023 877996 > 872 [i]