Best Known (233−30, 233, s)-Nets in Base 4
(233−30, 233, 69908)-Net over F4 — Constructive and digital
Digital (203, 233, 69908)-net over F4, using
- 44 times duplication [i] based on digital (199, 229, 69908)-net over F4, using
- net defined by OOA [i] based on linear OOA(4229, 69908, F4, 30, 30) (dual of [(69908, 30), 2097011, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(4229, 1048620, F4, 30) (dual of [1048620, 1048391, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4229, 1048624, F4, 30) (dual of [1048624, 1048395, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- linear OA(4221, 1048576, F4, 30) (dual of [1048576, 1048355, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(48, 48, F4, 4) (dual of [48, 40, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(48, 85, F4, 4) (dual of [85, 77, 5]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4229, 1048624, F4, 30) (dual of [1048624, 1048395, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(4229, 1048620, F4, 30) (dual of [1048620, 1048391, 31]-code), using
- net defined by OOA [i] based on linear OOA(4229, 69908, F4, 30, 30) (dual of [(69908, 30), 2097011, 31]-NRT-code), using
(233−30, 233, 515496)-Net over F4 — Digital
Digital (203, 233, 515496)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4233, 515496, F4, 2, 30) (dual of [(515496, 2), 1030759, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4233, 524314, F4, 2, 30) (dual of [(524314, 2), 1048395, 31]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4229, 524312, F4, 2, 30) (dual of [(524312, 2), 1048395, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4229, 1048624, F4, 30) (dual of [1048624, 1048395, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- linear OA(4221, 1048576, F4, 30) (dual of [1048576, 1048355, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(48, 48, F4, 4) (dual of [48, 40, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(48, 85, F4, 4) (dual of [85, 77, 5]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- OOA 2-folding [i] based on linear OA(4229, 1048624, F4, 30) (dual of [1048624, 1048395, 31]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4229, 524312, F4, 2, 30) (dual of [(524312, 2), 1048395, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4233, 524314, F4, 2, 30) (dual of [(524314, 2), 1048395, 31]-NRT-code), using
(233−30, 233, large)-Net in Base 4 — Upper bound on s
There is no (203, 233, large)-net in base 4, because
- 28 times m-reduction [i] would yield (203, 205, large)-net in base 4, but