Best Known (32−9, 32, s)-Nets in Base 2
(32−9, 32, 58)-Net over F2 — Constructive and digital
Digital (23, 32, 58)-net over F2, using
- 21 times duplication [i] based on digital (22, 31, 58)-net over F2, using
- net defined by OOA [i] based on linear OOA(231, 58, F2, 9, 9) (dual of [(58, 9), 491, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(231, 58, F2, 8, 9) (dual of [(58, 8), 433, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(210, 32, F2, 8, 4) (dual of [(32, 8), 246, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (6, 10, 32)-net over F2, using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 5]-NRT-code), using
- linear OOA(221, 29, F2, 8, 9) (dual of [(29, 8), 211, 10]-NRT-code), using
- extracting embedded OOA [i] based on digital (12, 21, 29)-net over F2, using
- linear OOA(210, 32, F2, 8, 4) (dual of [(32, 8), 246, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(231, 58, F2, 8, 9) (dual of [(58, 8), 433, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(231, 58, F2, 9, 9) (dual of [(58, 9), 491, 10]-NRT-code), using
(32−9, 32, 86)-Net over F2 — Digital
Digital (23, 32, 86)-net over F2, using
- net defined by OOA [i] based on linear OOA(232, 86, F2, 9, 9) (dual of [(86, 9), 742, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(232, 86, F2, 8, 9) (dual of [(86, 8), 656, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(232, 86, F2, 2, 9) (dual of [(86, 2), 140, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(232, 172, F2, 9) (dual of [172, 140, 10]-code), using
- adding a parity check bit [i] based on linear OA(231, 171, F2, 8) (dual of [171, 140, 9]-code), using
- a “Gra†code from Grassl’s database [i]
- adding a parity check bit [i] based on linear OA(231, 171, F2, 8) (dual of [171, 140, 9]-code), using
- OOA 2-folding [i] based on linear OA(232, 172, F2, 9) (dual of [172, 140, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(232, 86, F2, 2, 9) (dual of [(86, 2), 140, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(232, 86, F2, 8, 9) (dual of [(86, 8), 656, 10]-NRT-code), using
(32−9, 32, 470)-Net in Base 2 — Upper bound on s
There is no (23, 32, 471)-net in base 2, because
- 1 times m-reduction [i] would yield (23, 31, 471)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 2147 554331 > 231 [i]