Best Known (210−45, 210, s)-Nets in Base 4
(210−45, 210, 1056)-Net over F4 — Constructive and digital
Digital (165, 210, 1056)-net over F4, using
- 42 times duplication [i] based on digital (163, 208, 1056)-net over F4, using
- trace code for nets [i] based on digital (7, 52, 264)-net over F256, using
- net from sequence [i] based on digital (7, 263)-sequence over F256, using
- trace code for nets [i] based on digital (7, 52, 264)-net over F256, using
(210−45, 210, 4373)-Net over F4 — Digital
Digital (165, 210, 4373)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4210, 4373, F4, 45) (dual of [4373, 4163, 46]-code), using
- 266 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0, 1, 29 times 0, 1, 45 times 0, 1, 64 times 0, 1, 86 times 0) [i] based on linear OA(4199, 4096, F4, 45) (dual of [4096, 3897, 46]-code), using
- an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- 266 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0, 1, 29 times 0, 1, 45 times 0, 1, 64 times 0, 1, 86 times 0) [i] based on linear OA(4199, 4096, F4, 45) (dual of [4096, 3897, 46]-code), using
(210−45, 210, 1582329)-Net in Base 4 — Upper bound on s
There is no (165, 210, 1582330)-net in base 4, because
- 1 times m-reduction [i] would yield (165, 209, 1582330)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 676930 463302 244140 949843 169057 695491 255671 147206 583725 326080 495484 767277 246026 959923 028536 874540 805085 198589 529059 450011 252756 > 4209 [i]