Best Known (53, 53+32, s)-Nets in Base 9
(53, 53+32, 344)-Net over F9 — Constructive and digital
Digital (53, 85, 344)-net over F9, using
- 7 times m-reduction [i] based on digital (53, 92, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 46, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 46, 172)-net over F81, using
(53, 53+32, 691)-Net over F9 — Digital
Digital (53, 85, 691)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(985, 691, F9, 32) (dual of [691, 606, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(985, 728, F9, 32) (dual of [728, 643, 33]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,31], and designed minimum distance d ≥ |I|+1 = 33 [i]
- discarding factors / shortening the dual code based on linear OA(985, 728, F9, 32) (dual of [728, 643, 33]-code), using
(53, 53+32, 99728)-Net in Base 9 — Upper bound on s
There is no (53, 85, 99729)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 1290 106305 335787 597527 981108 689254 943700 356548 690116 619031 172967 042755 273320 191617 > 985 [i]