Best Known (177, 177+31, s)-Nets in Base 4
(177, 177+31, 17476)-Net over F4 — Constructive and digital
Digital (177, 208, 17476)-net over F4, using
- net defined by OOA [i] based on linear OOA(4208, 17476, F4, 31, 31) (dual of [(17476, 31), 541548, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4208, 262141, F4, 31) (dual of [262141, 261933, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4208, 262144, F4, 31) (dual of [262144, 261936, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(4208, 262144, F4, 31) (dual of [262144, 261936, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4208, 262141, F4, 31) (dual of [262141, 261933, 32]-code), using
(177, 177+31, 101211)-Net over F4 — Digital
Digital (177, 208, 101211)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4208, 101211, F4, 2, 31) (dual of [(101211, 2), 202214, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4208, 131076, F4, 2, 31) (dual of [(131076, 2), 261944, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4208, 262152, F4, 31) (dual of [262152, 261944, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4208, 262153, F4, 31) (dual of [262153, 261945, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(29) [i] based on
- linear OA(4208, 262144, F4, 31) (dual of [262144, 261936, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(40, 9, F4, 0) (dual of [9, 9, 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(30) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(4208, 262153, F4, 31) (dual of [262153, 261945, 32]-code), using
- OOA 2-folding [i] based on linear OA(4208, 262152, F4, 31) (dual of [262152, 261944, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(4208, 131076, F4, 2, 31) (dual of [(131076, 2), 261944, 32]-NRT-code), using
(177, 177+31, large)-Net in Base 4 — Upper bound on s
There is no (177, 208, large)-net in base 4, because
- 29 times m-reduction [i] would yield (177, 179, large)-net in base 4, but