Free Bit Index Calculator for Polar Codes

An expert tool for determining information and frozen bit indices in polar codes, based on the principles used in MATLAB’s 5G Toolbox.

Calculation Results

Free Bit Indices (K)

Frozen Bit Indices (N-K)

What is a Free Bit Index Calculation for Polar Codes?

A free bit index calculation for polar codes is the process of determining which synthetic sub-channels are reliable enough to carry user data (information bits, or “free bits”) and which are too noisy and must carry fixed, known data (frozen bits). This selection is the fundamental step in polar code construction, a concept introduced by Erdal Arıkan. The core idea, known as channel polarization, is to transform a physical communication channel into a set of N virtual sub-channels with polarized reliabilities: some become nearly perfect (noiseless), while others become completely useless (pure noise).

This calculator simulates the process, similar to what one might perform in an environment like MATLAB using its 5G Toolbox functions (e.g., `nrPolarEncode`), to identify these two sets of indices. The K most reliable indices become the ‘free bits,’ designated for carrying the message, while the remaining N-K indices are ‘frozen,’ typically to a value of 0.

Polar Code Formula and Explanation

There isn’t a single, simple formula for the free bit index calculation. Instead, it’s an algorithmic process based on estimating the reliability of each of the N synthetic channels. The most common method, especially in academic contexts, is based on the Bhattacharyya parameter, Z(W), which measures the reliability of a channel W. A lower Z value means a more reliable channel (closer to 0 is noiseless), while a higher Z value means a less reliable channel (closer to 1 is pure noise).

The calculation proceeds as follows:

1. Initial Parameter: Calculate the initial Bhattacharyya parameter Z for the underlying physical channel (e.g., an AWGN channel) based on the Design SNR. For an AWGN channel, a common approximation is Z ≈ e^-SNR, where SNR is the linear signal-to-noise ratio (not in dB).
2. Recursive Calculation: The reliabilities of the N synthetic channels are calculated recursively. For a step from N/2 channels to N channels, the Bhattacharyya parameters (let’s call them ‘z_vals’) are updated using the rule:
   • For the first N/2 new channels: z_new = 2*z_old – z_old²
   • For the second N/2 new channels: z_new = z_old²

   This process is repeated log2(N) times.

3. Sorting: After the final iteration, there will be N Bhattacharyya parameter values, one for each synthetic sub-channel. These values are then sorted from lowest (most reliable) to highest (least reliable).
4. Selection: The indices corresponding to the K lowest Bhattacharyya values are selected as the free bit indices. The remaining N-K indices are the frozen bit indices.

For more advanced implementations, check out our guide on Successive Cancellation List (SCL) Decoding.

Variables Table

Key Variables in Polar Code Construction

Variable	Meaning	Unit	Typical Range
N	Codeword Length	Unitless Integer	32, 64, 128, 256, 512, 1024 (must be a power of 2)
K	Message Length	Unitless Integer	1 to N-1
Design SNR	Signal-to-Noise Ratio	dB (decibels)	-5 dB to 20 dB
Z(W)	Bhattacharyya Parameter	Unitless Probability	0 (perfect channel) to 1 (useless channel)

Practical Examples

Example 1: High-Rate Code

Imagine a scenario with good channel conditions where we want to send a lot of data relative to the block size.

• Inputs: N = 256, K = 200, Design SNR = 5.0 dB
• Process: The algorithm calculates the reliability for all 256 sub-channels at an SNR of 5.0 dB. It then selects the indices of the 200 most reliable channels.
• Results: The calculator would output a list of 200 free bit indices and 56 frozen bit indices. The free bit indices would be those that correspond to the channels that polarized towards being ‘noiseless’.

Example 2: Low-Rate Code (Robust)

Now, consider a noisy environment where we need high robustness. We sacrifice data rate for reliability.

• Inputs: N = 256, K = 50, Design SNR = -2.0 dB
• Process: The algorithm runs again, but at a lower SNR, the overall reliability of all channels drops. It selects only the 50 absolute best channels.
• Results: The calculator provides a list of 50 free bit indices and a much larger list of 206 frozen bit indices. This creates a very robust code, as data is only placed on the most resilient virtual channels. Learn more about rate matching in 5G NR.

How to Use This Free Bit Index Calculator

Using this calculator is a straightforward process for anyone familiar with channel coding concepts.

1. Select Codeword Length (N): Choose the total block size for your polar code from the dropdown. This is always a power of 2.
2. Enter Message Length (K): Input the number of information bits you intend to send. This determines the code rate (R = K/N).
3. Set Design SNR: Enter the channel Signal-to-Noise ratio in dB. This is a critical parameter that dictates the reliability ordering. A lower SNR will result in a different set of reliable indices compared to a high SNR.
4. Calculate and Interpret: Click “Calculate Indices”. The tool will display the number of free and frozen bits, along with the specific zero-based indices for each set. The accompanying chart visualizes the reliability of all N channels, making it easy to see the polarization effect.

Key Factors That Affect Free Bit Index Calculation

• Design SNR: This is the most influential factor. The set of reliable channels is highly dependent on the assumed SNR. A code constructed for 0 dB will have a different reliability order than one constructed for 10 dB.
• Codeword Length (N): The polarization effect becomes more pronounced as N increases. With a larger N, the sub-channels polarize more extremely to either very good or very bad, making the selection clearer.
• Construction Algorithm: While Bhattacharyya parameter-based construction is common, other methods like Density Evolution (DE) or Gaussian Approximation (GA) exist and can yield slightly different reliability orders. This calculator uses a Bhattacharyya-based approach suitable for AWGN channels.
• Systematic vs. Non-Systematic Codes: This calculator assumes a non-systematic construction, which is the direct output of the polarization process. Systematic codes involve an extra transformation step.
• Kernel Choice: The standard polar code uses a 2×2 kernel matrix. Variations in this kernel can change the polarization behavior, a topic of ongoing research. Explore our article on advanced polar code construction for details.
• Rate-Matching: In practical systems like 5G, the output of the polar encoder is often punctured or repeated to match a specific transmission block size. This rate-matching process relies on the initial reliability order. Our 5G polar coding tutorial explains this in depth.

Frequently Asked Questions (FAQ)

1. Why are the indices different from what my textbook shows?

The exact reliability order depends on the construction method (Bhattacharyya, GA, etc.) and the Design SNR. This calculator uses a standard Bhattacharyya approximation for AWGN, which is a common but not the only method. Many textbooks use simplified examples for a Binary Erasure Channel (BEC), which produces a fixed reliability order regardless of SNR.

2. What does “free bit” mean?

“Free bit” is another term for an information bit. It’s a position in the pre-encoded block `u` that is “free” to be filled with the data you want to transmit.

3. Why are some bits “frozen”?

Bits are frozen on the most unreliable channels. Sending actual information on them would lead to a very high probability of error. By fixing their value (usually to 0), the decoder has a known reference point, which helps it correctly decode the other bits during the successive cancellation process.

4. How does this relate to MATLAB’s `nrPolarEncode` function?

The 3GPP standard for 5G specifies a fixed reliability sequence calculated offline. MATLAB’s `nrPolarEncode` function uses this pre-calculated sequence. This calculator performs a live calculation based on the Bhattacharyya parameter to demonstrate the underlying principle of how such a sequence is generated. The core concept is identical: select the K most reliable channels for information.

5. Why must N be a power of 2?

The channel polarization process is based on a recursive splitting of channels into two. Starting with a single channel, you get 2, then 4, then 8, and so on. This recursive binary structure naturally requires the final block length N to be a power of 2.

6. Can I use this for a Binary Erasure Channel (BEC)?

This calculator is specifically tuned for an AWGN channel model via the SNR input. For a BEC with erasure probability ‘p’, the Bhattacharyya parameter is simply ‘p’, and the recursive formulas are the same. However, the initial Z value would be different.

7. What happens if K is very close to N?

If K is close to N (a high-rate code), you will be forced to use some less reliable channels to carry information. This makes the code more susceptible to errors, and it will require a higher SNR to perform well.

8. Where can I learn more about the decoding process?

The standard decoding algorithm is called Successive Cancellation (SC) Decoding. For better performance, Successive Cancellation List (SCL) decoding is often used. We have a great resource on polar decoding algorithms.

Related Tools and Internal Resources

Expand your knowledge of channel coding and 5G with our other expert tools and articles:

• LDPC Code Generator: Explore the other major channel coding scheme used in 5G for data channels.
• Shannon Capacity Calculator: Understand the theoretical maximum data rate for a given bandwidth and SNR.
• 5G NR Throughput Calculator: Estimate practical data rates in a 5G New Radio system.