Best Known (68, 89, s)-Nets in Base 4
(68, 89, 1032)-Net over F4 — Constructive and digital
Digital (68, 89, 1032)-net over F4, using
- 41 times duplication [i] based on digital (67, 88, 1032)-net over F4, using
- trace code for nets [i] based on digital (1, 22, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 22, 258)-net over F256, using
(68, 89, 1340)-Net over F4 — Digital
Digital (68, 89, 1340)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(489, 1340, F4, 21) (dual of [1340, 1251, 22]-code), using
- 303 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 10 times 0, 1, 17 times 0, 1, 25 times 0, 1, 37 times 0, 1, 51 times 0, 1, 64 times 0, 1, 78 times 0) [i] based on linear OA(476, 1024, F4, 21) (dual of [1024, 948, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 303 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 10 times 0, 1, 17 times 0, 1, 25 times 0, 1, 37 times 0, 1, 51 times 0, 1, 64 times 0, 1, 78 times 0) [i] based on linear OA(476, 1024, F4, 21) (dual of [1024, 948, 22]-code), using
(68, 89, 299896)-Net in Base 4 — Upper bound on s
There is no (68, 89, 299897)-net in base 4, because
- 1 times m-reduction [i] would yield (68, 88, 299897)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 95782 057780 466523 818428 745089 597437 287467 125746 982376 > 488 [i]