Best Known (127, 127+26, s)-Nets in Base 4
(127, 127+26, 5041)-Net over F4 — Constructive and digital
Digital (127, 153, 5041)-net over F4, using
- net defined by OOA [i] based on linear OOA(4153, 5041, F4, 26, 26) (dual of [(5041, 26), 130913, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(4153, 65533, F4, 26) (dual of [65533, 65380, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4153, 65536, F4, 26) (dual of [65536, 65383, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(4153, 65536, F4, 26) (dual of [65536, 65383, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(4153, 65533, F4, 26) (dual of [65533, 65380, 27]-code), using
(127, 127+26, 28165)-Net over F4 — Digital
Digital (127, 153, 28165)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4153, 28165, F4, 2, 26) (dual of [(28165, 2), 56177, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4153, 32772, F4, 2, 26) (dual of [(32772, 2), 65391, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4153, 65544, F4, 26) (dual of [65544, 65391, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(4153, 65536, F4, 26) (dual of [65536, 65383, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4145, 65536, F4, 25) (dual of [65536, 65391, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 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(25) ⊂ Ce(24) [i] based on
- OOA 2-folding [i] based on linear OA(4153, 65544, F4, 26) (dual of [65544, 65391, 27]-code), using
- discarding factors / shortening the dual code based on linear OOA(4153, 32772, F4, 2, 26) (dual of [(32772, 2), 65391, 27]-NRT-code), using
(127, 127+26, large)-Net in Base 4 — Upper bound on s
There is no (127, 153, large)-net in base 4, because
- 24 times m-reduction [i] would yield (127, 129, large)-net in base 4, but