Best Known (87−15, 87, s)-Nets in Base 7
(87−15, 87, 117650)-Net over F7 — Constructive and digital
Digital (72, 87, 117650)-net over F7, using
- 71 times duplication [i] based on digital (71, 86, 117650)-net over F7, using
- net defined by OOA [i] based on linear OOA(786, 117650, F7, 15, 15) (dual of [(117650, 15), 1764664, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(786, 823551, F7, 15) (dual of [823551, 823465, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(785, 823543, F7, 15) (dual of [823543, 823458, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(778, 823543, F7, 13) (dual of [823543, 823465, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(71, 8, F7, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(786, 823551, F7, 15) (dual of [823551, 823465, 16]-code), using
- net defined by OOA [i] based on linear OOA(786, 117650, F7, 15, 15) (dual of [(117650, 15), 1764664, 16]-NRT-code), using
(87−15, 87, 411776)-Net over F7 — Digital
Digital (72, 87, 411776)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(787, 411776, F7, 2, 15) (dual of [(411776, 2), 823465, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(787, 823552, F7, 15) (dual of [823552, 823465, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(787, 823553, F7, 15) (dual of [823553, 823466, 16]-code), using
- construction XX applied to Ce(14) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- linear OA(785, 823543, F7, 15) (dual of [823543, 823458, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(778, 823543, F7, 13) (dual of [823543, 823465, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(771, 823543, F7, 12) (dual of [823543, 823472, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(71, 9, F7, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(14) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(787, 823553, F7, 15) (dual of [823553, 823466, 16]-code), using
- OOA 2-folding [i] based on linear OA(787, 823552, F7, 15) (dual of [823552, 823465, 16]-code), using
(87−15, 87, large)-Net in Base 7 — Upper bound on s
There is no (72, 87, large)-net in base 7, because
- 13 times m-reduction [i] would yield (72, 74, large)-net in base 7, but