# Background: Finite Length Bounds

To evaluate the performance of error correcting codes in the finite blocklength regime, the traditional asymptotic information theoretic results are not very insightful. Instead, finite blocklength coding bounds provide better means to assess
their performance.

Here, we provide the implementation of three different bounds for the scalar AWGN channel:

- Shannon's 1959 Sphere Packing Bound
- Gallager's Random Coding Bound
- Normal Approximation

To interpret the results of the Online Tool, please consider the following points:

- The dropdown menu
**Bound type** chooses the bound to plot. For both the Random Coding Bound (RCB) and the Normal Approximation (NA), three parameters have be specified: The constellation size, the transmission rate and the
blockength. For the sphere packing bound (SP), the transmission rate and the blocklength suffice.
**Constellation size**: The RCB and the NA assume discrete complex signaling with an M-QAM constellation. The value of M is one of 4, 16, 64, 256.
**Transmission rate**: The transmission rate is bounded between 0 and log_{2}(M) bits. Its unit is information bits/complex channel use.
**Blocklength**: This value refers to the number of *complex* channel uses.

## Sphere Packing Bound

Detailed description will follow.

## Random Coding Bound

Detailed description will follow.

## Normal Approximation

Detailed description will follow.