"What do you think this is?"
Mira opened an old notebook and showed a cryptic sequence of numbers.
"0110100111010011..." Yuki read aloud. "Some kind of code?"
Aoi approached to look. "Might be communication code. Mira, where did you find this?"
Mira quietly pointed. An old communication engineering book in the library.
"To decode this, we first need to understand the nature of the channel," Aoi drew a diagram on the whiteboard.
"Channel?" Yuki asked.
"The path between sender and receiver through which messages travel. Real communication channels always have noise."
Mira wrote a note. "Binary Symmetric Channel, p = 0.1"
"Binary symmetric channel," Aoi explained. "A model where each bit flips with probability p. If p=0.1, there's a 10 percent chance of 0 becoming 1 or 1 becoming 0."
Yuki thought. "So the sent message and received message might be different."
"Yes. How accurately can information be sent through noise? That's measured by channel capacity."
Aoi wrote an equation.
"C = 1 - H(p)
Where H(p) is the entropy of the bit flip probability"
"Sounds difficult..." Yuki frowned.
"Let's calculate concretely. For p=0.1, H(0.1) ≈ 0.47 bits"
"So channel capacity is 1 - 0.47 = 0.53 bits?"
"Exactly! This means on this channel, per bit transmitted, only an average of 0.53 bits of information reliably gets through."
Mira nodded and showed another note. "Shannon limit"
"Shannon limit," Aoi's voice grew excited. "An amazing theorem Shannon proved. Below channel capacity rate, error probability can approach arbitrarily close to zero."
"How?" Yuki leaned forward.
"Proper encoding. By cleverly adding redundancy to information, you can recover the original message even with noise."
Aoi showed an example.
"For instance, if channel capacity is 0.5 bits/symbol, use 2 bits to send 1 bit of information. The remaining 1 bit is for error correction."
"But what encoding is optimal?"
"That's the difficult problem. Even knowing it's theoretically possible, finding concrete codes is hard."
Mira opened another page of the notebook. Complex matrices and bit patterns.
"Hamming codes, Reed-Solomon codes, turbo codes," Aoi pointed. "These are practical codes that approach the Shannon limit."
Yuki looked back at the original sequence. "So this sequence is also some encoded message?"
Mira quietly nodded.
Aoi began analysis. "Let's look for patterns. Any repetition? Periodicity?"
The three wrote in their notebooks. Dividing the bit string, grouping, taking statistics.
"Ah, when divided into 7-bit groups, I can see a pattern," Yuki noticed.
"Maybe Hamming(7,4) code," Aoi confirmed. "A code adding 3 parity bits to 4 information bits."
Mira smiled. An unusual expression for her.
Aoi began decoding. Parity check, error correction, extraction of original information bits.
"Done."
In the notebook, the decoded message.
"Information is the resolution of uncertainty"
Yuki read aloud. "Information is the resolution of uncertainty."
"Shannon's words," Aoi said with deep feeling.
Mira left a new note. "Communication is the art of reliable transmission over unreliable channels"
"The art of reliable transmission over unreliable channels," Yuki translated.
Rain began falling outside the window. In a noisy world, there are still messages that reliably get through. Today, they learned that.