Trellis-oriented G-codes

The following tables depicts properties of some distance 4, 6, and 8 lexicodes and trellis-oriented G-codes. Over a binary field, lexicodes with odd minimum distance are simply punctured lexicodes with even minimum distance, so it is redundant to list their properties [14].

The minimum distance 4 lexicodes are all extended Hamming codes or shortenings thereof [14]. The minimum distance 6 and 8 codes listed also have optimal error-correction capability in the sense that each has the best known minimum distance for its length and dimension as compared to [8]. We can see that the trellis-oriented codes almost always have lower decoding complexity than the corresponding lexicodes.

Dim- Lexicodes Trellis- Lexicodes Trellis- Lexicodes Trellis-

Length Length States States 2| E| - | V| + 1 2| E| - | V| + 1

2 6 2 19

3 7 3 39

4 8 3 55

5 10 3 69

6 11 3 107

7 12 3 123

8 13 4 235

9 14 4 259

10 15 4 331

11 16 4 355

12 18 4 369

13 19 4 407

14 20 4 423

15 21 4 563

16 22 4 587

17 23 4 659

18 24 4 683

19 25 5 1,219

20 26 5 1,243

21 27 5 1,315

22 28 5 1,339

23 29 5 1,603

24 30 5 1,627

25 31 5 1,699

26 32 5 1,723

27 34 5 1,737

28 35 5 1,775

29 36 5 1,791

30 37 5 1,931

31 38 5 1,955

32 39 5 2,027

33 40 5 2,051

34 41 5 2,643

35 42 5 2,667

36 43 5 2,739

37 44 5 2,763

38 45 5 3,027

39 46 5 3,051

40 47 5 3,123

41 48 5 3,147

42 49 6 5,491

43 50 6 5,515

44 51 6 5,587

45 52 6 5,611

46 53 6 5,875

47 54 6 5,899

48 55 6 5,971

49 56 6 5,995

50 57 6 7,027

51 58 6 7,051

52 59 6 7,123

53 60 6 7,147

54 61 6 7,411

55 62 6 7,435

56 63 6 7,507

57 64 6 7,531

58 66 6 7,545

59 67 6 7,583

60 68 6 7,599

61 69 6 7,739

62 70 6 7,763

63 71 6 7,835

64 72 6 7,859

65 73 6 8,451

66 74 6 8,475

67 75 6 8,547

68 76 6 8,571

69 77 6 8,835

70 78 6 8,859

71 79 6 8,931

72 80 6 8,955

73 81 6 11,411

74 82 6 11,435

75 83 6 11,507

76 84 6 11,531

77 85 6 11,795

78 86 6 11,819

79 87 6 11,891

80 88 6 11,915

81 89 6 12,947

82 90 6 12,971

83 91 6 13,043

84 92 6 13,067

85 93 6 13,331

86 94 6 13,355

87 95 6 13,427

88 96 6 13,451

89 97 7 23,251

90 98 7 23,275

91 99 7 23,347

92 100 7 23,371

93 101 7 23,635

94 102 7 23,659

95 103 7 23,731

96 104 7 23,755

97 105 7 24,787

98 106 7 24,811

99 107 7 24,883

100 108 7 24,907

101 109 7 25,171

102 110 7 25,195

103 111 7 25,267

104 112 7 25,291

105 113 7 29,395

106 114 7 29,419

107 115 7 29,491

108 116 7 29,515

109 117 7 29,779

110 118 7 29,803

111 119 7 29,875

112 120 7 29,899

113 121 7 30,931

114 122 7 30,955

115 123 7 31,027

116 124 7 31,051

117 125 7 31,315

118 126 7 31,339

119 127 7 31,411

120 128 7 31,435

Dim- Lexicode Trellis- Lexicode Trellis- Lexicode Trellis-

Length Length States States 2| E| - | V| + 1 2| E| - | V| + 1

2 9 2 27

3 11 3 55

4 12 4 111

5 14 4 181 141

6 15 5 4 335 215

7 16 5 451 387

8 17 6 747

9 18 7 6 1,275 955

10 20 8 6 1,821 1,053

11 21 8 6 2,655 1,279

12 22 8 7 3,555 2,083

13 24 8 7 4,333 2,117

14 25 9 7 6,143 2,351

15 26 9 7 7,715 3,011

16 27 9 8 9,323 5,035

17 28 9 8 10,235 5,435

18 29 9 12,827 10,651

19 30 31 10 9 14,939 10,701

20 32 10 9 18,141 10,943

21 33 10 9 19,167 11,683

22 34 10 9 23,523 12,267

23 35 10 9 25,067 13,947

24 36 10 9 29,691 18,075

25 37 10 31,259 27,739

26 38 11 44,635 47,835

27 39 11 53,979 54,235

28 41 11 55,005 54,285

29 42 11 62,431 54,527

30 43 11 71,139 55,203

31 44 11 77,291 55,403

32 45 11 83,451 58,107

33 46 11 89,627 64,027

34 47 11 95,835 71,771

35 48 11 102,107 92,891

36 49 12 11 144,347 93,275

37 50 12 11 157,147 96,987

38 51 12 170,459 146,395

39 52 53 12 178,651 146,493

40 53 54 12 190,939 146,783

41 55 12 203,229 147,555

42 56 12 215,519 150,251

43 57 12 221,667 151,035

44 58 12 240,107 151,835

45 59 12 252,411 157,275

46 60 12 253,979 172,763

47 61 12 267,867 204,763

48 62 12 277,211 206,811

49 63 12 302,043 275,931

50 64 12 308,699 277,467

51 65 12 328,155 279,003

52 66 12 336,347 299,483

53 67 12 348,635 301,019

54 68 13 541,147 488,923

55 69 13 565,723 498,139

56 70 71 13 571,867 498,189

57 72 13 602,589 498,431

58 73 13 639,455 499,107

59 74 13 664,035 500,843

60 75 13 688,619 505,083

61 76 13 700,923 505,883

62 77 13 704,027 516,699

63 78 13 704,859 518,363

64 79 13 707,291 526,299

65 80 13 750,555 560,603

66 81 13 787,931 564,699

67 82 13 791,003 668,123

68 83 13 819,675 671,195

69 84 13 14 834,011 913,883

70 85 13 14 930,267 916,955

71 86 13 14 954,843 938,459

72 87 13 14 979,419 956,891

73 88 14 1,487,323 1,030,619

74 89 14 1,490,395 1,055,195

75 90 91 14 1,518,043 1,055,293

76 91 92 14 1,548,763 1,055,583

77 92 93 14 1,659,355 1,056,355

78 93 94 14 1,683,931 1,056,459

79 95 14 1,782,237 1,058,491

80 96 14 1,806,815 1,059,483

81 97 14 1,880,547 1,066,843

82 98 14 1,929,707 1,067,355

83 99 14 1,978,875 1,079,515

84 100 14 2,003,483 1,109,979

85 101 14 2,077,275 1,121,755

86 102 14 2,101,979 1,127,899

87 103 14 2,108,379 1,212,891

88 104 14 2,139,611 1,225,179

89 105 14 2,177,499 1,419,739

90 106 14 2,183,643 1,976,795

91 107 14 2,216,411 1,979,867

92 108 14 2,228,699 1,989,083

93 109 14 2,429,403 2,013,659

94 110 14 2,511,323 2,025,947

95 111 14 2,560,475 2,038,235

96 112 14 15 2,609,627 3,607,003

97 113 14 15 2,658,779 3,613,147

98 114 15 4,067,803 3,631,579

99 115 15 4,092,379 3,828,187

100 116 15 4,215,259 3,831,259

Dim- Standard Trellis- Standard Trellis- Standard Trellis-

Length Length States States 2| E| - | V| + 1 2| E| - | V| + 1

2 12 2 35

3 14 3 71

4 15 4 143

5 16 4 195

6 18 5 341

7 19 6 647

8 20 6 779

9 21 7 1,547

10 22 8 2,395

11 23 9 4,219

12 24 9 4,475

13 28 9 4,529

14 30 9 4,777

15 31 9 5,463

16 32 9 5,515

17 34 9 7,645 5,693

18 35 9 12,671 6,143

19 36 9 12,803 6,275

20 37 10 9 24,267 7,691

21 38 10 9 25,115 8,539

22 39 10 9 26,939 10,363

23 40 10 9 27,195 10,619

24 42 10 31,805 17,853

25 43 11 41,791 33,087

26 44 11 41,987 33,283

27 45 12 65,227

28 46 12 67,163

29 47 12 72,571 78,203

30 48 12 72,827 78,459

31 49 50 13 12 135,611 80,317

32 50 51 13 12 169,019 87,103

33 51 52 13 12 248,123 88,643

34 52 53 13 248,507 137,547

35 53 54 14 13 427,579 138,331

36 54 55 14 13 431,419 142,203

37 55 56 14 13 442,171 142,459

38 56 57 14 442,555 274,875

39 58 14 487,997 308,283

40 59 14 628,031 457,019

41 60 14 628,227 460,091

42 62 14 629,197 460,861

43 63 14 631,903 464,703

44 64 14 632,035 464,899

45 65 15 14 1,263,819 581,835

46 66 15 1,287,195 1,053,275

47 67 15 16 1,346,235 2,106,299

48 68 69 15 16 1,701,883 2,106,557

50 71 15 16 1,704,575 2,115,203

51 72 15 16 2,041,731 2,232,203

52 73 15 16 2,126,219 2,539,931

53 74 16 3,835,291 3,351,995

54 75 16 17 3,976,635 6,123,003

55 76 77 16 17 4,849,147 6,123,389

56 77 78 17 7,937,659 6,125,439