LFSR Maximal Period Calculator

Compute maximum period of a Linear-Feedback Shift Register (LFSR) and list maximal-length tap polynomials for common bit widths.

Calculator Electronics Updated Apr 23, 2026
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How to Use
  1. Enter number of shift register bits (width).
  2. Tool returns maximal period 2^N - 1 and a standard maximum-length polynomial.
Input
bits (2-32)
Hz (kHz, MHz OK)
Presets
LFSR Info
Max Period
—
Taps (Fibonacci)
—
Polynomial
—
Repeat Time
—

Show Work

Enter bit width.

Max-Length Taps

Period
2^N − 1
Maximum cycle length.
7-bit PRBS-7
Taps 7,6 · period 127
Ethernet test.
9-bit PRBS-9
Taps 9,5 · period 511
USB test.
15-bit PRBS-15
Taps 15,14 · period 32767
SDH/SONET.
23-bit PRBS-23
Taps 23,18 · period 8.4M
ITU-T O.151.
31-bit PRBS-31
Taps 31,28 · period 2.1G
10GbE PRBS.

History of the LFSR

Solomon Golomb's 1967 book Shift Register Sequences codified the theory of linear-feedback shift register pseudo-random sequences, building on Marsh and Zierler's 1940s work on maximal-length polynomials. CCSDS standardized PRBS-7, PRBS-15, PRBS-23, and PRBS-31 for spacecraft telemetry in the 1970s. LFSRs became ubiquitous in consumer electronics as CRC generators (Ethernet's CRC-32 uses a 32-bit LFSR) and as PN-code generators in CDMA cellular and GPS.

About This Calculator

Enter register width N (2-32 bits). The tool returns maximal period 2^N − 1, a standard max-length tap pair (from the Zierler-Brillhart tables), and the corresponding polynomial in hex.

LFSRs are everywhere in digital systems: CRC computation, pseudo-random test patterns (PRBS-7 through PRBS-31), spread-spectrum chipping codes, and legacy hardware PRNGs. For cryptographic randomness, LFSRs alone are too predictable — use cryptographic PRNGs (ChaCha20, AES-CTR-DRBG). Everything runs client-side.

Frequently Asked Questions

What is an LFSR?

Linear-Feedback Shift Register: a shift register with XOR feedback taps. Produces pseudo-random binary sequences. Core of CRC generators, pseudo-noise sequences for spread-spectrum, and simple hardware PRNGs.

Maximal length?

An N-bit LFSR produces at most 2^N - 1 unique states before repeating (the all-zero state is excluded). A maximal-length tap polynomial hits all 2^N - 1 states in a single cycle.

Uses?

Pseudo-noise sequence generation (GPS C/A code uses 10-bit LFSR giving 1023-chip sequence). CRC code generation. Simple hardware random-number sources. Bit-error-rate test pattern generation.

Common Use Cases

GPS C/A Code

10-bit LFSR × 2, XOR gives 1023-chip Gold code per satellite.

PRBS Tester

PRBS-7 (2^7 − 1 = 127 bit pattern) for Ethernet BER test.

CRC-32

32-bit LFSR with polynomial 0x04C11DB7 — Ethernet CRC.

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