On Vector Random Linear Network Coding in Wireless Broadcasts.

Area of Science:

• Information theory
• Wireless communication networks
• Coding theory

Background:

• Scalar linear network coding (LNC) over finite fields GF(2L) is well-studied.
• Random LNC (RLNC) shows potential for wireless broadcast delay improvement.
• Prior work primarily focused on scalar RLNC, leaving vector RLNC performance unexplored.

Purpose of the Study:

• To analyze the completion delay performance of primitive vector RLNC over GF(2L).
• To derive theoretical expressions for completion delay distributions and expected values.
• To establish a performance baseline for future vector RLNC schemes.

Main Methods:

• Derivation of closed-form expressions for completion delay probability distribution.
• Calculation of expected completion delay at single receivers and system-wide.
• Theoretical analysis of performance limitations and asymptotic behavior.
• Validation through numerical simulations.

Main Results:

• Primitive vector RLNC over GF(2L) inherently fails to achieve optimal completion delay, unlike scalar RLNC.
• The gap between expected and optimal completion delay at a receiver is a constant (<0.714).
• Expected completion delay, normalized by packet number P, is asymptotically optimal as P increases.

Conclusions:

• Primitive vector RLNC provides a performance baseline for practical vector RLNC designs.
• Vector LNC offers flexibility but has inherent limitations compared to scalar LNC regarding optimal delay.
• The study provides a theoretical foundation for understanding and optimizing vector RLNC in wireless networks.