Best Known (153−20, 153, s)-Nets in Base 4
(153−20, 153, 104859)-Net over F4 — Constructive and digital
Digital (133, 153, 104859)-net over F4, using
- 1 times m-reduction [i] based on digital (133, 154, 104859)-net over F4, using
- net defined by OOA [i] based on linear OOA(4154, 104859, F4, 21, 21) (dual of [(104859, 21), 2201885, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4154, 1048591, F4, 21) (dual of [1048591, 1048437, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4154, 1048597, F4, 21) (dual of [1048597, 1048443, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4154, 1048597, F4, 21) (dual of [1048597, 1048443, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4154, 1048591, F4, 21) (dual of [1048591, 1048437, 22]-code), using
- net defined by OOA [i] based on linear OOA(4154, 104859, F4, 21, 21) (dual of [(104859, 21), 2201885, 22]-NRT-code), using
(153−20, 153, 524299)-Net over F4 — Digital
Digital (133, 153, 524299)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4153, 524299, F4, 2, 20) (dual of [(524299, 2), 1048445, 21]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4152, 524299, F4, 2, 20) (dual of [(524299, 2), 1048446, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4152, 1048598, F4, 20) (dual of [1048598, 1048446, 21]-code), using
- construction X4 applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(421, 22, F4, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,4)), using
- dual of repetition code with length 22 [i]
- linear OA(41, 22, F4, 1) (dual of [22, 21, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(20) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(4152, 1048598, F4, 20) (dual of [1048598, 1048446, 21]-code), using
- 41 times duplication [i] based on linear OOA(4152, 524299, F4, 2, 20) (dual of [(524299, 2), 1048446, 21]-NRT-code), using
(153−20, 153, large)-Net in Base 4 — Upper bound on s
There is no (133, 153, large)-net in base 4, because
- 18 times m-reduction [i] would yield (133, 135, large)-net in base 4, but