Best Known (219−32, 219, s)-Nets in Base 4
(219−32, 219, 16385)-Net over F4 — Constructive and digital
Digital (187, 219, 16385)-net over F4, using
- 1 times m-reduction [i] based on digital (187, 220, 16385)-net over F4, using
- net defined by OOA [i] based on linear OOA(4220, 16385, F4, 33, 33) (dual of [(16385, 33), 540485, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4220, 262161, F4, 33) (dual of [262161, 261941, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4220, 262165, F4, 33) (dual of [262165, 261945, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(29) [i] based on
- linear OA(4217, 262144, F4, 33) (dual of [262144, 261927, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4199, 262144, F4, 30) (dual of [262144, 261945, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(32) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(4220, 262165, F4, 33) (dual of [262165, 261945, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4220, 262161, F4, 33) (dual of [262161, 261941, 34]-code), using
- net defined by OOA [i] based on linear OOA(4220, 16385, F4, 33, 33) (dual of [(16385, 33), 540485, 34]-NRT-code), using
(219−32, 219, 124448)-Net over F4 — Digital
Digital (187, 219, 124448)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4219, 124448, F4, 2, 32) (dual of [(124448, 2), 248677, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4219, 131082, F4, 2, 32) (dual of [(131082, 2), 261945, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4219, 262164, F4, 32) (dual of [262164, 261945, 33]-code), using
- 1 times truncation [i] based on linear OA(4220, 262165, F4, 33) (dual of [262165, 261945, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(29) [i] based on
- linear OA(4217, 262144, F4, 33) (dual of [262144, 261927, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4199, 262144, F4, 30) (dual of [262144, 261945, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(32) ⊂ Ce(29) [i] based on
- 1 times truncation [i] based on linear OA(4220, 262165, F4, 33) (dual of [262165, 261945, 34]-code), using
- OOA 2-folding [i] based on linear OA(4219, 262164, F4, 32) (dual of [262164, 261945, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(4219, 131082, F4, 2, 32) (dual of [(131082, 2), 261945, 33]-NRT-code), using
(219−32, 219, large)-Net in Base 4 — Upper bound on s
There is no (187, 219, large)-net in base 4, because
- 30 times m-reduction [i] would yield (187, 189, large)-net in base 4, but