MinT - Best Known (144−16, 144, s)-Nets in Base 8

Best Known (144−16, 144, s)-Nets in Base 8

(144−16, 144, 2097358)-Net over F₈ — Constructive and digital

Digital (128, 144, 2097358)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 22, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 11, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
2. digital (106, 122, 2097150)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹²², 2097150, F₈, 18, 16) (dual of [(2097150, 18), 37748578, 17]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OOA(8¹²², 8388601, F₈, 2, 16) (dual of [(8388601, 2), 16777080, 17]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8¹²², 8388602, F₈, 2, 16) (dual of [(8388602, 2), 16777082, 17]-NRT-code), using
        trace code [i] based on linear OOA(64⁶¹, 4194301, F₆₄, 2, 16) (dual of [(4194301, 2), 8388541, 17]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁶¹, 8388602, F₆₄, 16) (dual of [8388602, 8388541, 17]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁶¹, large, F₆₄, 16) (dual of [large, large−61, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(144−16, 144, 2097664)-Net in Base 8 — Constructive

(128, 144, 2097664)-net in base 8, using

base change [i] based on digital (92, 108, 2097664)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (8, 16, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 8, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
  2. digital (76, 92, 2097150)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16⁹², 2097150, F₁₆, 18, 16) (dual of [(2097150, 18), 37748608, 17]-NRT-code), using
      - OOA 4-folding and stacking with additional row [i] based on linear OOA(16⁹², 8388601, F₁₆, 2, 16) (dual of [(8388601, 2), 16777110, 17]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁹², 8388602, F₁₆, 2, 16) (dual of [(8388602, 2), 16777112, 17]-NRT-code), using
        trace code [i] based on linear OOA(256⁴⁶, 4194301, F₂₅₆, 2, 16) (dual of [(4194301, 2), 8388556, 17]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256⁴⁶, 8388602, F₂₅₆, 16) (dual of [8388602, 8388556, 17]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁶, large, F₂₅₆, 16) (dual of [large, large−46, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(144−16, 144, large)-Net over F₈ — Digital

Digital (128, 144, large)-net over F₈, using

t-expansion [i] based on digital (125, 144, large)-net over F₈, using
- 2 times m-reduction [i] based on digital (125, 146, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁴⁶, large, F₈, 21) (dual of [large, large−146, 22]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(8¹⁴⁵, large, F₈, 21) (dual of [large, large−145, 22]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(144−16, 144, large)-Net in Base 8 — Upper bound on s

There is no (128, 144, large)-net in base 8, because

14 times m-reduction [i] would yield (128, 130, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (144−16, 144, s)-Nets in Base 8

(144−16, 144, 2097358)-Net over F8 — Constructive and digital

(144−16, 144, 2097664)-Net in Base 8 — Constructive

(144−16, 144, large)-Net over F8 — Digital

(144−16, 144, large)-Net in Base 8 — Upper bound on s

(144−16, 144, 2097358)-Net over F₈ — Constructive and digital

(144−16, 144, large)-Net over F₈ — Digital