 Open Access
 Total Downloads : 234
 Authors : Venkata Krishna Rao M
 Paper ID : IJERTV5IS050714
 Volume & Issue : Volume 05, Issue 05 (May 2016)
 DOI : http://dx.doi.org/10.17577/IJERTV5IS050714
 Published (First Online): 23052016
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
on the Periodicity of NonMaximal Length Linear Feedback Shift Register Sequences
Venkata Krishna Rao M,
Professor and Head,
Department of Electronics & Communication Engineering, Vidya Jyothi Institute of Technology,
Hyderabad, India
AbstractPseudoNoise (PN) sequences are widely used in, from simple applications such as clock dividers to complex applications such as spread spectrum communication systems. Among PN sequences, maximal length (m)sequences are very popular for communications and ranging applications, due to their desirable properties. While msequences are available only in limited number of lengths, the nonmaximal length (NML) sequences are available in varieties of lengths. It appears that the NMLS were not investigated into, to the extent they deserve. The author conjectures that several other (non communications) applications might benefit from the NLM sequences. In this paper, generation and periodicity property of NLM sequences, corresponding to polynomials of degree 3 to 20 are investigated. The results are expected to create an interest among the researchers in exploring some new applications of NML sequences.
Keywords PN sequences, Reducible polynomials, Linear feedback shift register sequences, MobiusMu function, EulerPhi function

INTRODUCTION
PseudoNoise (PN) sequences find applications in spread spectrum communication systems [1], radar [2], clock dividers [3], system identification [4,5,6,7,8], VLSI circuit testing [9,10], testpattern generation and for signature analysis [11], scrambling in digital broadcasting and communications [12], cryptography [13] and programmable sound generators [14]. A special class of PN sequences,
primitive irreducible polynomials and their connection to m sequences is also discussed. Section III is dedicated to Non maximal Length Shift Register (NMLSR) sequences. In this section, generation and periodicity property of NMLSR sequences corresponding to polynomials of degree 3 to 20 are investigated analytically. Section IV discusses the details of simulations and the results. Conclusions and scope of future work are presented in section V.

BINARY LINEAR FEEDBACK SHIFT REGISTER SEQUENCES
Let a binary polynomial h(x) of degree n is given by
() = 0 + 11 + + 1 + (1)
where h0 = hn = 1 and other coefficients {0,1}. The polynomial h(x) can also be represented as a binary vector
= (0, 1, , 1, ) expressed either in binary or in octal notation. The vector h or equivalently the polynomial h(x) can be used to generate a binary sequence =
0, 1, 2, using an nstage binary shift register circuit with a feedback tap connected to ith stage if hi = 1 for 0 <
. Since hn = 1, the nth stage always has a connection. Thus h(x) is said to be the generator polynomial of the sequence .
called the Binary Linear Feedback Shift Register (LFSR)
=
+
+ +
(2)
Sequences are generated using a simple hardware i.e. shift
register whose individual stage outputs are feedback to the
1 1
2 2
input through a modulo2 adder. The LFSRs with certain combinations of feedback connections give rise to sequences of maximum length i.e. 2 1, where n is the number of stages of the shift register. These sequences are called maximal sequences (msequences), which are preferred to nonmaximal sequences in most of the real systems, due to their desirable properties. Accordingly, the nonmaximal sequences were almost ignored and their properties were not explicitly discussed in the literature. The author conjectures that several other applications might benefit from non maximal length sequences.
In this paper, the nonmaximal length sequences generated by linear feedback shift register and their periodicity property are discussed in detail. The paper is organized as follows. In section I, the mathematical structure associated with LFSR sequence generator is discussed. The
A 5stage linear feedback shift register circuit corresponding to the polynomial x5+ x3+1 is shown in Fig.1. If the current output is taken from the kth stage, then (k1)st is the previous stage, (k2)nd is the second previous stage and so on. Now by dividing the given polynomial by x5, we get the polynomial 1+ x2+ x5. Here a positive power means an advancement of a bit position, where as a negative power means a delay of the bit position. Thus both the polynomials represent the same shift register circuit. The former takes the reference stage at the right, while the latter considers the reference stage at the left. The binary vector corresponding to the polynomial x5+ x3+1 is (101001) which is 51 in octal notation.
The sequence a generated by the LFSR circuit initially loaded with a nonzero binary vector in Fig. 1 repeats every N bits. If the polynomial is a primitive irreducible, then the sequence length is maximized with a period of = 2 1,
where n is the number of stages of the shift register which is same as the degree of the generator polynomial . This sequence is popularly known as maximal length sequence (MLS) or msequence [15]. The sequences generated with different initial shift register states (i.e. initial content of shift register) are the same except for a cyclic shift. If the polynomial is reducible into factors, then < 2 1 and the sequences are called nonmaximal length sequences (NMLS). The variable L is used to represent the length of NMLS and
< .
ak
we get () = 1288 = 16. i.e. there are 16 primitive polynomials of degree 8. This means out of 30 irreducible polynomials of degree 8, only 16 polynomials are primitive which give rise to msequences of length 255. The remaining
14 polynomials are not primitive and generate nonmaximal length sequences having lengths < 255. Table I gives the number of nonmaximal length sequences as computed from
(3) and (4). It may noted that for degrees 3,4,5,7,13,17 and 19, no nonmaximal length sequences exist meaning that all irreducible polynomials are primitive only.
TABLE I. UMBER OF NONMAXIMAL LENGTH SEQUENCES FOR DEGREES
Degree
n
Number of irreducible polynomials
()
Number of maximal length sequences
()
Number of nonmaximal length sequences
() ()
3
2
2
0
4
2
2
0
5
6
6
0
6
9
6
3
7
18
18
0
8
30
16
14
9
56
48
8
10
99
60
39
11
186
176
10
12
335
144
191
13
630
630
0
14
1161
756
405
15
2182
1800
382
16
4080
2048
2032
17
7710
/td>
7710
0
18
14532
7776
6756
19
27594
27594
0
20
52377
24000
28377
Degree
n
Number of irreducible polynomials
()
Number of maximal length sequences
()
Number of nonmaximal length sequences
() ()
3
2
2
0
4
2
2
0
5
6
6
0
6
9
6
3
7
18
18
0
8
30
16
14
9
56
48
8
10
99
60
39
11
186
176
10
12
335
144
191
13
630
630
0
14
1161
756
405
15
2182
1800
382
16
4080
2048
2032
17
7710
7710
0
18
14532
7776
6756
19
27594
27594
0
20
52377
24000
28377
k x5(1)
k1 k2 k3
x3(x2)
k4
1(x5)
3 TO 20
Fig. 1. A 5stage linear feedback shift register with tap connections corresponding to the polynomial x5+ x3+1 or 1+ x2+ x5.
The msequences [15,16] have some desirable properties which make them unanimously preferred over msequences. The most important property is 2level thumbtack autocorrelation function and extremely low cross correlation function of the set of sequences corresponding to the generator polynomial of same degree. Whereas the msequences are available in limited number of lengths i.e. = 2 1, the NMLS are available in varieties of lengths.
For a given degree n, a polynomial must be irreducible if it has to generate either an msequence or a nonmaximal length sequence (nmsequence). Only the irreducible polynomials that are primitive generate msequences. For a given degree, the number of either msequences or nmsequences available depends on the number of the primitive and nonprimitive polynomials available. The number of such polynomials can be found using MÃ¶bius function and Euler function respectively as follows.

Number of Irreducible Polynomials
The number of irreducible polynomials modulo2 of degree n is given by
() = 1 2 () (3)
/
where (. ) is the MÃ¶bius function [15] the sum is over all positive divisors of d of n. For n= 8, the divisors of 8 are
= 1,2,4,8, then = 8, 4, 2, 1 and (8) = 0,0, 1, 1. Substituting in (3), we get (8) = 30 i.e. there are 30 irreducible polynomials of degree 8. These are the LFSR sequences that can be generated using a shift register of 8 stages.

Number of Primitive Polynomials
The number of primitive polynomials modulo2 of degree
n is given by
(2 1)
() = (4)
where (. ) is the Euler function [15]. For n= 8,
(255) = (3.5.17) = 2.4.16 = 128. Substituting in (4),


NONMAXMIAL LENGTH SHIFT REGISTER SEQUENCES In this section, the lengths of nonmaximal length
sequences for degrees 3 to 20 are computed. If the number
= 2 1 has factors other than 1 and N itself, then both primitive and nonprimitive polynomials exist for the degree n.
Table II gives the lengths of maximal length sequences L for degrees 3 to 20. The third column gives the factors of L. These factors are nothing but the lengths of nonmaximal length sequences. These factors (equivalently, the sequence lengths) are used as arguments in Euler function to compute the number of sequences having these lengths. The maximal lengths N corresponding to n=3,5,7,13,17 and 19 i.e. 7, 31, 127, 8191, 131071 and 524287 are called mersenne primes.
Let us consider = 8, then = 255 which has factors: 1, 3, 5, 15, 17, 51, 85 and 255. The sequences generated by the irreducible polynomials of degree 8 must have one of the periods: 1, 3, 5, 15, 17, 51, 85 and 255. The case of 15 is ignored as it is already considered for lower degree n=4, N=15. As 3 and 5 are factors of 15, they are also ignored. There are 30 irreducible polynomials (please refer to Table I) which give rise to sequences of periods: 17, 51, 85 and 255. There are 16 msequences having length of 255
corresponding to 16 primitive polynomials. The remaining 14 sequences have lengths: 17, 51 and 85. The number of sequences having these lengths can be again found from (4). Thus (17) = 16, (51) = 32 and (85) = 64, giving rise to (. ) = 168 = 2, (. ) = 328 = 4 and (. ) = 648 = 8. Thus there are two sequences of length 17, four sequences of length 51 and eight sequences of length 85, thus making a total of 14 nonmaximal length sequences (please refer to Table III).
Similarly, for degree 6, = 26 1 = 63 has factors
sorted in ascending order: 1, 3, 7, 9, 21, 63. Out of these the lengths of 3 and 7 were already obtained from lower degrees 2 and 3 respectively. Since (9)6 = 66 =1, there is one sequence of length 9 and since (21)6 = 126 = 2, there are two sequences of length 21. Similarly,
(63)6 = 366 = 6, there are six sequences of length 63, which are msequences. Thus the total number of sequences obtained for degree 6 are 9, out of which three sequences are of nonmaximal length. Samples of the actual sequences obtained through simulations are given in section IV.
TABLE II. MAXIMAL LENGTHS (L) AND FACTORS OF L FOR DEGREE 6 TO 20
Degree
n
= 2 1
Factors of
3
7
1.7
4
15
1.3.5.15
5
31
1.31 (Mersenne Prime)
6
63
1.3.7.9.21.63
7
127
1.127 (Mersenne Prime)
8
255
1.3.5.15.17.51.85.255
9
511
1.7.73.511
10
1023
1.3.11.31.33.93.341.1023
11
2047
1.23.89.2047
12
4095
1.3.5.7.9.13.15.21.35.39.45.63.65.91.105.117.
195.273.315.455.585.819.1365.4095
13
8191
1.8191 (Mersenne Prime)
14
16383
1.3.43.127.129.381.5461.16383
15
32767
1.7.31.151.217.1057.4681.32767
16
65535
1.3.5.15.17.51.85.255.257.771.1285.3855.
4369.13107.21845.65535
17
131071
1.131071 (Mersenne Prime)
18
262143
1.3.7.9.19.21.27.57.63.73.13.171.189.219.
399.511.513.657.1197.1387.1533.1971.3591.
4161.4599.9709.12483.13797.29127.37449.
87381.262143
19
524287
1.524287 (Mersenne Prime)
20
1048575
1.3.5.11.15.25.31.33.41.55.75.93.123.155.165.
205.275.341.451.465.615.775.825.1023.1025.
1271.1353.1705.2255.2325.3075.3813.5115.
6355.6765.8525.11275.13981.19065.25575.
31775.33825.41943.69905.95325.209715.
349525.1048575
The number of nonmaximal length sequences of different lengths obtained for a given degree n are computed and listed in Table III for degrees 3 to 19. The sequence lengths for degree 20 are avoided due to space constraints, but can be computed from factors listed in Table II and using (4).
TABLE III. LENGTHS OF NONMAXMIAL LENGTH SEQUENCES (NMLS) FOR DEGREE 6 TO 20
Degree n
Period of NMLS
Number of NMLS
Total Number of NMLS
6
21
2
9
1
3
8
85
8
51
4
17
2
14
9
73
8
8
10
341
30
93
6
33
2
11
1
39
11
89
8
23
2
10
12
1365
48
819
36
585
24
455
24
315
12
273
12
195
8
117
6
105
4
91
6
65
4
45
2
39
2
35
2
13
1
191
14
5461
378
381
18
129
6
43
3
405
15
4681
300
1057
60
217
12
151
10
382
16
21845
1024
13107
512
4369
256
3855
128
1285
64
771
32
257
16
2032
18
87381
2592
37449
1296
29127
864
13797
432
12483
432
9709
432
4599
144
4161
144
3591
108
1971
72
1533
48
1387
72
1197
36
657
24
513
18
399
12
219
8
189
6
171
6
133
6
57
2
27
1
19
1
6756
2017 [10 3 2 1 0]
341
2257 [10 7 5 3 2 1 0]
341
2065 [10 5 4 2 0]
93
2653 [10 8 7 5 3 1 0]
341
3753 [10 9 8 7 6 5 3 1 0]
341
3573 [10 9 8 6 5 4 3 1 0]
341
3043 [10 9 5 1 0]
33
2107 [10 6 2 1 0]
341
3061 [10 9 5 4 0]
341
2547 [10 8 6 5 2 1 0]
341
3453 [10 9 8 5 3 1 0]
93
3121 [10 9 6 4 0]
341
2701 [10 8 7 6 0]
341
2437 [10 8 4 3 2 1 0]
341
2413 [10 8 3 1 0]
93
2311 [10 7 6 3 0]
341
3777 [10 9 8 7 6 5 4 3 2 1 0]
11
3607 [10 9 8 7 2 1 0]
341
2355 [10 7 6 5 3 2 0]
341
10
3315 [10 9 7 6 3 2 0]
3601 [10 9 8 7 0]
341
341
3651 [10 9 8 7 5 3 0]
341
2541 [10 8 6 5 0]
93
3255 [10 9 7 5 3 2 0]
341
3277 [10 9 7 5 4 3 2 1 0]
341
3367 [10 9 7 6 5 4 2 1 0]
341
3421 [10 9 8 4 0]
341
2143 [10 6 5 1 0]
341
3465 [10 9 8 5 4 2 0]
341
3247 [10 9 7 5 2 1 0]
93
2123 [10 6 4 1 0]
341
2035 [10 4 3 2 0]
341
3705 [10 9 8 7 6 2 0]
341
3205 [10 9 7 2 0]
93
2231 [10 7 4 3 0]
341
3777 [10 9 8 7 6 5 4 3 2 1 0]
11
3417 [10 9 8 3 2 1 0]
341
2671 [10 8 7 5 4 3 0]
341
2251 [10 7 5 3 0]
33
2633 [10 8 7 4 3 1 0]
341
2017 [10 3 2 1 0]
341
2257 [10 7 5 3 2 1 0]
341
2065 [10 5 4 2 0]
93
2653 [10 8 7 5 3 1 0]
341
3753 [10 9 8 7 6 5 3 1 0]
341
3573 [10 9 8 6 5 4 3 1 0]
341
3043 [10 9 5 1 0]
33
2107 [10 6 2 1 0]
341
3061 [10 9 5 4 0]
341
2547 [10 8 6 5 2 1 0]
341
3453 [10 9 8 5 3 1 0]
93
3121 [10 9 6 4 0]
341
2701 [10 8 7 6 0]
341
2437 [10 8 4 3 2 1 0]
341
2413 [10 8 3 1 0]
93
2311 [10 7 6 3 0]
341
3777 [10 9 8 7 6 5 4 3 2 1 0]
11
3607 [10 9 8 7 2 1 0]
341
2355 [10 7 6 5 3 2 0]
341
10
3315 [10 9 7 6 3 2 0]
3601 [10 9 8 7 0]
341
341
3651 [10 9 8 7 5 3 0]
341
2541 [10 8 6 5 0]
93
3255 [10 9 7 5 3 2 0]
341
3277 [10 9 7 5 4 3 2 1 0]
341
3367 [10 9 7 6 5 4 2 1 0]
341
3421 [10 9 8 4 0]
341
2143 [10 6 5 1 0]
341
3465 [10 9 8 5 4 2 0]
341
3247 [10 9 7 5 2 1 0]
93
2123 [10 6 4 1 0]
341
2035 [10 4 3 2 0]
341
3705 [10 9 8 7 6 2 0]
341
3205 [10 9 7 2 0]
93
2231 [10 7 4 3 0]
341
3777 [10 9 8 7 6 5 4 3 2 1 0]
11
3417 [10 9 8 3 2 1 0]
341
2671 [10 8 7 5 4 3 0]
341
2251 [10 7 5 3 0]
33
2633 [10 8 7 4 3 1 0]
341

SIMULATIONS AND RESULTS
Simulations are carried out to find out the lengths of the nonmaximal length sequences corresponding to polynomials of degree 6 to 20. All the polynomials of degrees 6 to 20 listed in [17] were considered in the simulation study. The polynomials were represented in octal notation. The octal string of each polynomial is parsed into individual octal digits, then each digit is converted to a binary number. The binary vector = (0, 1, , 1, ) thus obtained is used to decide the feedback tap connections as described in section II.
Customized Matlab programs are developed for converting octal notation to feedback taps and then generating the binary linear feedback shift register (LFSR) sequences recursively. The recursive loop is continued to get a sequence a of 3.25N binary digits, accounting for more than 4 – 8 cycles of non maximal length sequence depending the actual period of the sequence. The value of 3.25 is not mandatory, but is used for the unambiguous computation of autocorrelation peaks and subsequently the sequence period.
The cyclic correlation of the sequence is computed and a peak detection algorithm is applied to find out the correlation peaks, the sample count between the peaks is taken as the repetition period of the sequence. The measured periods , thus obtained for different polynomials in octal notation are listed in the third column of Table IV. For this work, in total
around 20 customized functions in Matlab were developed for the polynomial representation, sequence generation, period computation and displaying of the results. All these functions are called in main program i.e. PeriodityofNMLSequences.m.
Degree
Polynomial Octal [Taps]
Period
127 [6 4 2 1 0]
21
6
111 [6 3 0]
9
165 [6 5 4 2 0]
21
567 [8 6 5 4 2 1 0]
85
763 [8 7 6 5 4 1 0]
51
675 [8 7 5 4 3 2 0]
85
727 [8 7 6 4 2 1 0]
17
613 [8 7 3 1 0]
85
433 [8 4 3 1 0]
51
8
477 [8 5 4 3 2 1 0]
735 [8 7 6 4 3 2 0]
85
85
637 [8 7 4 3 2 1 0]
51
573 [8 6 5 4 3 1 0]
85
643 [8 7 5 1 0]
85
661 [8 7 5 4 0]
51
771 [8 7 6 5 4 3 0]
85
471 [8 5 4 3 0]
17
1231 [9 7 4 3 0]
73
1027 [9 4 2 1 0]
73
1401 [9 8 0]
73
9
1511 [9 8 6 3 0]
1145 [9 6 5 2 0]
73
73
1641 [9 8 7 5 0]
73
1003 [9 1 0]
73
1113 [9 6 3 1 0]
73
Degree
Polynomial Octal [Taps]
Period
127 [6 4 2 1 0]
21
6
111 [6 3 0]
9
165 [6 5 4 2 0]
21
567 [8 6 5 4 2 1 0]
85
763 [8 7 6 5 4 1 0]
51
675 [8 7 5 4 3 2 0]
85
727 [8 7 6 4 2 1 0]
17
613 [8 7 3 1 0]
85
433 [8 4 3 1 0]
51
8
477 [8 5 4 3 2 1 0]
735 [8 7 6 4 3 2 0]
85
85
637 [8 7 4 3 2 1 0]
51
573 [8 6 5 4 3 1 0]
85
643 [8 7 5 1 0]
85
661 [8 7 5 4 0]
51
771 [8 7 6 5 4 3 0]
85
471 [8 5 4 3 0]
17
1231 [9 7 4 3 0]
73
1027 [9 4 2 1 0]
73
1401 [9 8 0]
73
9
1511 [9 8 6 3 0]
1145 [9 6 5 2 0]
73
73
1641 [9 8 7 5 0]
73
1003 [9 1 0]
73
1113 [9 6 3 1 0]
73
TABLE IV. POLYNOMIALS [TAPS] & PERIODS MEASURED OF NONMAXMIAL LENGTH SEQUENCES FOR DEGREE 6 TO 10
In addition to the above functions, 10 more Matlab programs are developed to compute the number of irreducible polynomials and the number of primitive polynomials through MÃ¶bius function and Euler function respectively. The results are used to populate the Tables I and II.
I the simulations study a total of 9840 (i.e. sum of values in the fourth column of Table III) sequences are generated, the measured periods from simulations are compared to the theoretical periods computed analytically using (3) and (4). Both the values are exactly same for all the 9840 sequences generated. To get an idea about the distribution of 1s and 0s visually, sample binary NML sequences a1 through a8 of degree 6 (period: 21) and degree 8 (period: 17, 51 and 85) along with the associated polynomials in octal form are displayed in Table V.
n=6 &
Polynomial1= 127 (Octal)
L= 21
a1=111111001110001000110
Polynomial2= 165 (Octal)
a2=111111011000100011100
n=8 &
Polynomial1= 727 (Octal)
L= 17
a1=11111111011000110
Polynomial2= 471 (Octal)
a2=11111111000101000
n=8 &
Polynomial1= 763 (Octal)
L= 51
a1=1111111101111010100111001001001111110001100
10111010
Polynomial1= 433 (Octal)
a2=1111111100001001110010001001010001011011100
01000010
Polynomial1= 637 (Octal)
a3=1111111101011101001100011111100100100111001
01011110
Polynomial1= 661 (Octal)
a4=1111111101000010001110110100010100100010011
10010000
n=8 &
Polynomial1= 567 (Octal)
L= 85
a1=11111111001000110110100111001010000100010000
01100011100000100111100110101001100100110
Polynomial2= 675 (Octal)
a2=11111111010001110101010110000101100110010000
01101110111000000100101101000000011100100
Polynomial3= 613 (Octal)
a3=11111111010100100100011101011101111100010001
00101100000110001011011110111001111110010
Polynomial4= 477 (Octal)
a4=11111111000101100111100000101110100001010100
11011110110110000100011101110110010111110
Polynomial5= 735 (Octal)
a5=11111111011001001100101011001111001000001110
00110000010001000010100111001011011000100
Polynomial6= 573 (Octal)
a6=11111111001001110000000101101001000000111011
10110000010011001101000011010101011100010
Polynomial7= 643 (Octal)
a7=11111111010011111100111011110110100011000001
10100100010001111101110101110001001001010
Polynomial8= 771 (Octal)
a8=11111111011111010011011101110001000011011011
11011001010100001011101000001111001101000
n=6 &
Polynomial1= 127 (Octal)
L= 21
a1=111111001110001000110
Polynomial2= 165 (Octal)
a2=111111011000100011100
n=8 &
Polynomial1= 727 (Octal)
L= 17
a1=11111111011000110
Polynomial2= 471 (Octal)
a2=11111111000101000
n=8 &
Polynomial1= 763 (Octal)
L= 51
a1=1111111101111010100111001001001111110001100
10111010
Polynomial1= 433 (Octal)
a2=1111111100001001110010001001010001011011100
01000010
Polynomial1= 637 (Octal)
a3=1111111101011101001100011111100100100111001
01011110
Polynomial1= 661 (Octal)
a4=1111111101000010001110110100010100100010011
10010000
n=8 &
Polynomial1= 567 (Octal)
L= 85
a1=11111111001000110110100111001010000100010000
01100011100000100111100110101001100100110
Polynomial2= 675 (Octal)
a2=11111111010001110101010110000101100110010000
01101110111000000100101101000000011100100
Polynomial3= 613 (Octal)
a3=11111111010100100100011101011101111100010001
00101100000110001011011110111001111110010
Polynomial4= 477 (Octal)
a4=11111111000101100111100000101110100001010100
11011110110110000100011101110110010111110
Polynomial5= 735 (Octal)
a5=11111111011001001100101011001111001000001110
00110000010001000010100111001011011000100
Polynomial6= 573 (Octal)
a6=11111111001001110000000101101001000000111011
10110000010011001101000011010101011100010
Polynomial7= 643 (Octal)
a7=11111111010011111100111011110110100011000001
10100100010001111101110101110001001001010
Polynomial8= 771 (Octal)
a8=11111111011111010011011101110001000011011011
11011001010100001011101000001111001101000
TABLE V. SAMPLE NONMAXMIAL LENGTH SEQUENCES
irreducible polynomials are computed analytically for the degrees 3 to 20. Simulations are carried out to generate a total of 9840 NML sequences corresponding to polynomials of degree 3 to 20. The period of each simulated sequence is computed to verify the analytical results. The repetition periods of all the simulated sequences are found to be exactly same as those obtained analytically. Sample NML sequences are also given.
The author conjectures that several applications other than communications might benefit from nonmaximal length sequences. Some of such applications could be the sound synthesizers. Work is in progress by the author towards such an investigation.
Matlab programs are also developed to display the results in ASCII string notation in compact form useful to populate the Table IV and V.

CONLUSIONS AND FUTURE WORK
The main focus of this work is the investigation of periodicity of the nonmaximal length sequences generated by linear feedback shift registers. The paper discussed the mathematical structure associated with LFSR sequence generator. The number of primitive and nonprimitive
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