Best Known (57, 79, s)-Nets in Base 9
(57, 79, 740)-Net over F9 — Constructive and digital
Digital (57, 79, 740)-net over F9, using
- 3 times m-reduction [i] based on digital (57, 82, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 41, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 41, 370)-net over F81, using
(57, 79, 5456)-Net over F9 — Digital
Digital (57, 79, 5456)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(979, 5456, F9, 22) (dual of [5456, 5377, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(979, 6572, F9, 22) (dual of [6572, 6493, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(19) ⊂ Ce(18) [i] based on
- linear OA(977, 6561, F9, 22) (dual of [6561, 6484, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(969, 6561, F9, 20) (dual of [6561, 6492, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(965, 6561, F9, 19) (dual of [6561, 6496, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(91, 10, F9, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(21) ⊂ Ce(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(979, 6572, F9, 22) (dual of [6572, 6493, 23]-code), using
(57, 79, 4376427)-Net in Base 9 — Upper bound on s
There is no (57, 79, 4376428)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 2427 497353 726716 107293 991930 682067 316284 036474 127459 540798 802110 510960 437665 > 979 [i]