Best Known (220−47, 220, s)-Nets in Base 4
(220−47, 220, 1060)-Net over F4 — Constructive and digital
Digital (173, 220, 1060)-net over F4, using
- trace code for nets [i] based on digital (8, 55, 265)-net over F256, using
- net from sequence [i] based on digital (8, 264)-sequence over F256, using
(220−47, 220, 4572)-Net over F4 — Digital
Digital (173, 220, 4572)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4220, 4572, F4, 47) (dual of [4572, 4352, 48]-code), using
- 461 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 8 times 0, 1, 19 times 0, 1, 37 times 0, 1, 61 times 0, 1, 89 times 0, 1, 111 times 0, 1, 126 times 0) [i] based on linear OA(4211, 4102, F4, 47) (dual of [4102, 3891, 48]-code), using
- construction X applied to Ce(46) ⊂ Ce(45) [i] based on
- linear OA(4211, 4096, F4, 47) (dual of [4096, 3885, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(4205, 4096, F4, 46) (dual of [4096, 3891, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(46) ⊂ Ce(45) [i] based on
- 461 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 8 times 0, 1, 19 times 0, 1, 37 times 0, 1, 61 times 0, 1, 89 times 0, 1, 111 times 0, 1, 126 times 0) [i] based on linear OA(4211, 4102, F4, 47) (dual of [4102, 3891, 48]-code), using
(220−47, 220, 1698198)-Net in Base 4 — Upper bound on s
There is no (173, 220, 1698199)-net in base 4, because
- 1 times m-reduction [i] would yield (173, 219, 1698199)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 709809 135459 436132 409667 577192 537753 496997 530060 397361 050759 448014 339981 799829 981034 584868 000674 471003 350357 961725 591040 218630 609852 > 4219 [i]