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# Sequence generationV15.8.0

Toolbox: Pseudo-random sequence generator
 (0,...30) 0100000000000000000000000000000 Output sequence length Output sequence c(n) 10000010110011111000101111110011001011001001110100

Toolbox: Low-PAPR sequence generator
 Sequence length MZC Largest prime num NZC

### Pseudo-random sequence generation

Generic pseudo-random sequences are defined by a length-31 Gold sequence. The output sequence c(n) of length , where is defined by:

where =1600 and the first m-sequence shall be initialized with . The initialization of the second m-sequence, , is denoted by with the value depending on the application of the sequence.

What it means: is the i-th bit of for i=0,...,30.

### Low-PAPR sequence generation

The low-PAPR sequence is defined by a cyclic shift of a base sequence according to

where is the length of the sequence. Multiple sequences are defined from a single base sequence through different values of 𝛼 and 𝛿.

Base sequences are divided into groups, where is the group number and v is the base sequence number within the group, such that each group contains one base sequence (𝑣 = 0) of each length and two base sequences (𝑣 = 0,1) of each length . The definition of the base sequence depends on the sequence length .

What it means:

• Each base sequence length has 30 groups of sequences.
• For base sequence length ≤ 72, there is only one base sequence within each group, i.e., v=0.
• For base sequence length > 72, there are two sequences within each group, i.e., v = 0 or 1.
• Base sequence of length {6, 12, 18, 24} are generated from pre-defined tables of φ
• Low-PAPR sequence is generated by rotating the base sequence with phase α.

#### Base sequences of length 36 or larger

For , the base sequence is given by

where

The length is given by the largest prime number such that

#### Base sequences of length less than 36

For the base sequence is given by

where the value of is given by Tables 5.2.2.2-1 to 5.2.2.2-4.

For , the base sequence is given by

u 0 1 2 3 φ(n) for Mzc=6 -3 -1 3 3 -1 -3 -3 3 -1 -1 3 -3 -3 -3 -3 3 1 -3 1 1 1 3 -1 -3 1 1 1 -3 -1 3 -3 1 -1 -3 -3 -3 -3 1 3 -3 -3 -3 -3 -1 1 -3 1 -1 -3 -1 -3 1 -3 -3 -3 -3 1 -3 3 -3 -3 1 3 1 -3 -3 -3 -1 -3 1 1 -3 1 1 3 -1 -3 3 1 1 3 3 -1 3 1 1 1 -3 3 -1 1 1 1 -1 3 -3 -3 -1 -1 -1 3 -1 -3 -3 -1 1 -1 -3 -3 -3 -3 1 -3 -1 -3 1 1 -3 -1 -3 -3 3 -3 1 1 -3 -3 1 -3 -3 -3 -1 1 1 -3 3 1 3 1 1 -3 -3 1 -3 1 1 3 -1 3 3 1 1 -3 1 3 3 1 1 -1 -1 3 -1 1 1 -1 3 -1 -1 1 1 -1 3 -3 -1 1 1 -3 1 -1 -1
u 0 1 2 3 4 5 6 7 8 9 φ(n) for Mzc=12 -3 1 -3 -3 -3 3 -3 -1 1 1 1 -3 -3 3 1 -3 1 3 -1 -1 1 3 3 3 -3 3 3 1 -3 3 -1 1 3 -3 3 -3 -3 -3 -1 3 3 3 -3 3 -3 1 -1 -3 -3 -1 -1 1 3 1 1 -1 1 -1 -3 1 -3 -3 3 1 -3 -3 -3 -1 3 -1 1 3 1 -1 3 -1 -1 -1 -3 -1 1 1 1 -3 -1 -3 3 -1 -3 -3 -3 -1 1 -1 1 -3 -3 -1 3 1 -3 -1 -3 3 1 3 3 1 -3 -1 -1 -3 -3 -1 -3 3 1 3 -1 -3 -3 3 -3 3 3 -3 -1 -1 3 3 1 -3 -3 -1 -3 -1 -1 -3 3 3 -1 -1 1 -3 -3 -1 3 -3 -3 -1 -3 1 -1 -3 3 3 -3 1 -1 -1 3 3 -3 -1 -1 -3 -1 -3 1 3 -3 1 3 3 3 1 -1 1 -1 3 -3 1 3 -1 -1 -3 -3 -1 -1 3 1 -3 -1 -1 -1 -1 1 -3 -1 3 3 -1 -3 1 -1 1 1 -1 1 3 3 -1 -1 -3 1 -3 -3 1 3 3 -1 -1 -3 3 3 -3 3 -3 -3 -3 3 -3 -1 3 3 3 -1 -3 1 -3 3 1 3 1 3 -3 -1 1 3 1 -1 -3 -3 3 1 3 -3 1 1 1 1 3 -3 3 -3 3 3 3 -1 -3 -3 -1 -3 1 3 -3 3 -1 -3 3 -3 -1 3 3 3 -3 -1 -3 -3 -1 1 -3 1 3 3 3 -1 -3 3 3 -3 3 1 -1 3 3 -3 1 -1 1 -1 1 -1 1 3 -3 1 -1 1 -1 -1 -3 1 -1 -3 -3 3 3 3 -3 -1 1 -3 3 1 -3 1 -1 3 1 1 -1 -1 -1 1 3 -3 1 -3 3 -3 3 -3 -3 3 -1 -1 1 3 -3
u 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 φ(n) for Mzc=18 -1 3 -1 -3 3 1 -3 -1 3 -3 -1 -1 1 1 1 -1 -1 -1 3 -3 3 -1 1 3 -3 -1 -3 -3 -1 -3 3 1 -1 3 -3 3 -3 3 1 -1 -1 3 -3 -1 1 1 1 1 1 -1 3 -1 -3 -1 -3 -3 3 3 3 1 -3 1 3 3 1 -3 -3 3 -1 -3 -1 1 1 1 -1 -1 -3 -1 1 -3 -3 -3 1 -3 -1 -1 1 -1 3 1 3 -3 1 1 3 -1 1 -1 -1 -3 1 1 -1 3 3 -3 3 -1 -3 3 -1 1 3 1 -3 -1 1 1 -3 1 3 3 -1 -3 -3 -3 1 1 -3 3 3 1 3 -3 3 -1 1 1 -1 1 -3 -3 -1 3 -3 1 -3 -3 1 -3 -3 3 1 -3 -1 -3 -3 -3 -1 1 1 3 3 -1 3 1 -3 -3 -1 1 -3 -3 3 3 3 1 3 -3 3 -3 -3 -3 -3 1 -3 3 1 1 3 -3 -3 1 3 -1 3 -3 -3 3 -3 -3 3 3 3 -1 -1 -3 -1 -1 -1 3 1 -3 -3 -1 3 -1 -3 -1 -3 -3 1 1 -1 -3 -1 -3 -1 -1 3 3 -1 3 1 3 1 1 -3 -3 -3 -3 1 3 -3 3 3 1 -3 -1 3 -1 -3 1 -3 3 -1 -3 -1 -3 1 1 -3 -3 -1 -1 3 -3 1 3 1 1 3 1 -3 1 -3 3 3 -1 -3 -3 -1 -3 -3 3 -3 -1 1 3 -3 -1 -3 -1 -3 1 3 -3 -1 3 3 3 1 -1 -3 3 -1 -3 -3 -1 3 3 -1 3 -1 -3 -1 1 -1 -3 -1 -1 -1 3 3 1 -3 1 -3 -1 -1 3 1 -3 -3 -3 -1 -3 -3 1 1 1 -1 -1 3 3 3 -3 -1 -3 -1 3 -1 1 -1 -3 1 -3 -3 -1 3 3 -3 1 1 -3 1 1 3 -3 -1 -3 -1 3 -3 3 -1 -1 -1 -3 1 -3 -1 -3 3 3 -1 -3 1 -3 -3 -1 -3 -1 1 3 3 3 -3 -3 1 -1 -1 1 1 -3 -1 3 3 3 3 -1 3 1 3 1 3 -1 -3 1 -3 -3 -3 3 3 -1 1 -3 -1 3 1 1 3 3 3 -1 -1 1 -3 -1 -3 -1 -3 -3 -1 -3 1 1 1 -3 -3 3 -3 -3 1 -3 3 3 3 -1 3 1 1 -3 -3 -3 3 -3 -1 -1 -3 -1 -1 -3 1 -3 3 -1 -1 -3 3 3 -3 -1 3 -1 -1 -1 -3 -3 3 3 -3 1 3 -1 -3 1 -1 -3 3 -3 -1 -1 -1 3 -1 -3 1 -3 -3 -3 1 1 3 3 -3 3 3 -3 -1 3 -3 1 -3 3 1 -1 -1 -1 -1 1 -1 3 3 -3 -1 1 3 -1 3 -1
u 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 φ(n) for Mzc=24 -1 -3 3 -1 3 1 3 -1 1 -3 -1 -3 -1 1 3 -3 -1 -3 3 3 3 -3 -3 -3 -1 -3 3 1 1 -3 1 -3 -3 1 -3 -1 -1 3 -3 3 3 3 -3 1 3 3 -3 -3 -1 -3 -3 1 -1 -1 -3 1 3 -1 -3 -1 -1 -3 1 1 3 1 -3 -1 -1 3 -3 -3 1 -3 3 -1 -3 -1 3 3 1 -1 1 1 3 -3 -1 -3 -3 -3 -1 3 -3 -1 -3 -3 -1 3 -3 -3 -1 3 -1 -1 1 3 1 3 -1 -1 -3 1 3 1 -1 -3 1 -1 -3 -3 -3 -1 1 -3 -3 1 1 -3 3 -1 -1 -3 1 3 1 -1 -3 -1 -3 1 -3 -3 -3 -3 -3 3 1 3 -1 1 -3 1 -3 1 -1 -3 -1 -3 -3 -3 -3 -1 -1 -1 1 1 -3 -3 -3 1 3 -1 1 -1 3 -3 3 -1 -3 -1 -3 3 -1 -1 -1 -3 -1 -1 -3 3 3 -3 -3 1 -3 3 -1 -1 -1 -3 3 1 -1 -3 -1 1 3 -1 1 -1 1 -3 -3 -3 -3 -3 1 1 -1 -3 -1 1 1 -3 1 -1 1 -3 3 -3 -3 3 -1 -3 1 3 -3 1 -3 -3 -3 -3 -3 -1 3 -3 3 1 3 1 -3 -1 -1 -3 1 1 3 1 -1 -3 3 1 3 -3 -3 3 -1 3 1 -1 -1 -1 3 3 1 1 1 3 3 1 -3 -3 -1 1 -3 1 3 -3 3 -3 3 -1 -3 1 3 1 -1 -1 -3 -1 3 -3 3 -1 -1 3 3 -3 -3 3 -3 -3 -3 3 -1 3 -1 3 3 1 1 -3 1 3 -3 3 -3 -3 -1 1 3 -3 -1 -1 -3 -3 -3 1 -3 -1 -1 3 1 3 -3 1 -1 3 3 -1 -3 3 -3 -1 -1 -3 -3 -3 3 -3 -3 -1 -1 -3 1 -3 -3 -1 -1 3 -1 1 -1 3 1 -3 -1 3 1 1 -1 -1 -3 -3 -3 -3 1 -1 3 3 -3 -1 1 -1 -1 1 1 -1 -1 3 -3 1 -3 1 -1 -1 -1 -3 3 -1 3 -1 1 -3 1 1 -3 -3 3 -3 -1 -1 -1 -1 -1 -3 -3 -1 1 1 -3 -3 -3 1 -3 1 -3 -3 1 -3 1 -3 -3 -3 -3 -3 1 -3 -3 1 1 -3 1 1 -3 -3 -3 -3 3 3 1 -1 -1 -1 1 -3 -1 1 -1 3 -3 -1 -3 -1 -1 1 -3 3 -1 -3 -3 -3 -1 -1 -1 -3 1 -1 -3 -1 3 -3 1 -3 3 -3 3 3 1 -1 -1 1 -3 -3 3 -1 1 -1 3 -3 1 1 3 -1 -3 3 1 -3 3 -1 -1 -1 -1 1 -3 -3 -3 -3 -3 1 -3 3 -3 1 -3 3 1 -1 -3 -1 -3 -3 -3 -3 1 3 -1 1 3 3 3 -3 -3 -1 1 -3 -1 -1 1 1 1 3 3 -1 1 -1 1 -1 -1 -3 -3 -3 3 1 -1 -3 -3 3 -1 -3 -1 -1 -1 3 -1 -1 3 -3 -1 3 -3 3 -3 -1 3 1 1 -1 -3 -3 -3 1 -1 -3 -3 -1 1 -3 -1 -3 1 1 -1 1 1 3 3 3 -1 1 -1 1 -1 -3 -1 3 -1 -1 3 3 -1 -1 -1 3 -1 -3 1 3 1 1 -3 -3 -3 -1 -3 -1 -3 -3 3 -3 -3 -1 3 3 -3 -1 3 1 1 1 3 -1 3 -3 -1 3 -1 3 1 -1 -3 -3 -3 1 -3 1 -3 1 1 3 1 -3 -3 -1 1 3 -1 -3 3 1 -1 -3 -3 -3 -3 -3 3 -3 -1 1 3 -1 -1 -3 -1 3 -1 -3 -1 -3 3 -1 3 1 1 -3 3 -3 -3 -3