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Feb 26, 2023 100 likes •608 views

Optimum MDS convolutional codes over GF(2 m ) and their relation to the trace functi n ngela Barbero and yvind Ytrehus UVa, Simula@UiB, UiB 1 Problem setting Unicast transmission over the Internet (Memoryless) packet erasure

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Optimum MDS convolutional codes

  • ver GF(2m)

and their relation to the trace functi n

Ángela Barbero and Øyvind Ytrehus UVa, Simula@UiB, UiB

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Problem setting

  • Unicast transmission over the Internet

– (Memoryless) packet erasure channel, capacity "1 − 𝜁"

  • Solutions in the Internet:

– TCP uses ARQ

  • Problem: Long round trip time (RTT) ≈ 100’s ms

– The recovery delay of any ARQ system large – Rate loss due to inexact RTT estimation

– Delay of recovery

  • If no delay constraints: ARQ sufficient in many cases
  • Applications with delay constraints: : Multimedia, IoT control

applications, stock market applications, games

– Better: Erasure correcting codes

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Coding criteria

  • Code rate close to channel capacity???
  • (Low) probability of recovery failure

– Either decoding failure: erasure pattern covers a codeword – Or recovery delay exceeding tolerance of application

  • Recovery complexity: Systematic codes?

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Coding candidates

  • MDS, Reed-Solomon: Long delay
  • «Rateless» , fountain codes: Long delay
  • Convolutional codes: «good» column distance

profile

– Binary? – q-ary – Flexible rate – Random codes

  • What does «random» mean??
  • Good column distance profile should still apply

Unsuited for delay sensitive app’s

«Block codes are for boys, convolutional codes are for men» – J. Massey

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Convolutional codes for dummies

Block code: 𝑑 = 𝑣𝐻 = 𝑣1 ⋯ 𝑣𝑙 𝑕11 ⋯ 𝑕1𝑜 ⋮ ⋱ ⋮ 𝑕𝑙1 ⋯ 𝑕𝑙𝑜 Minimum distance = min 𝑥 𝑑 : 𝑑 ≠ 0 Convolutional code: 𝑑 = 𝑣𝐻 = 𝑣(0) 𝑣(1) ⋯ 𝐻0 𝐻1 ⋯ 𝐻𝑀 𝐻0 ⋱ ⋮ ⋯ 𝐻0 𝑑(0) = 𝑣(0)𝐻0 CDP= min 𝑥 𝑑 0 , 𝑥 𝑑 0 𝑑 1 , ⋯ ∶ 𝑑(0) ≠ 0 , 𝑑(1) = 𝑣(0)𝐻1 + 𝑣(1)𝐻0,…

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Convolutional codes and erasure recovery for dummies

If CDP is (2,3,4, … , D) then an erasure pattern

– of weight 𝑘 and – starting at block/time 1

will be recovered at time 𝑘 iff 𝑘 < D

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  • H. Gluesing-Luerssen, J. Rosenthal, and R. Smarandache,

Strongly-MDS Convolutional Codes, IEEE Trans on IT 52, 2006. – E.Gabidulin, 1989

  • 𝑟-ary convolutional codes with optimum column distance

profile – MDS-convolutional codes

  • J. Rosenthal, and R. Smarandache, 1998
  • cdp = (𝑜 − 𝑙 + 1,2(𝑜 − 𝑙) + 1, … , D),

– Existence of MDS code equivalent to existence of superregular matrices – Existing constructions require large field

Convolutional code approach

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Our convolutional code approach

  • Systematic
  • Over GF(2m)
  • High rate 𝑜−1

𝑜

  • MDS (CDP=(2,3,4, … , D, D, D, ...))

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Parity-check matrix of a convolutional code

H0 H1 H2

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H(0)

Parity-check matrix of a convolutional code

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H(1)

Parity-check matrix of a convolutional code

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H(2)

Parity-check matrix of a convolutional code

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Generator matrix of a convolutional code

G(2)H(2)T=(0)

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Proper minors and superregularity

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Proper minors and superregularity

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Proper minors and superregularity

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Proper minors and superregularity

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Proper minors and superregularity

1 1 1 𝛽2 𝛽 1 1 1 1 𝛽 1 1 1 1 𝛽 𝛽 1

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Our contributions

  • s-superregularity
  • Constructions of MDS codes with CDP=(2,3, D =4)
  • Efficient algorithm to search for MDS codes with

CDP=(2,3,4,…, D ), D ≥ 5

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Proper minors and s-superregularity

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Proper minors and s-superregularity

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Proper minors and s-superregularity

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Proper minors and s-superregularity

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Proper minors and s-superregularity

1 1 𝛽 1 1 1 𝛽 1 𝛽 1 1 1 1 1 𝛽 1 1 𝛽3 1 1 𝛽 1 1

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Superregularity and CDP

Known for k=1: Gluesing-Luerssen et al 2006, Gabidulin 1989

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Binary superregular matrices?

  • 1-superregularity
  • 1x1:

1 → 2,1 𝑐𝑚𝑝𝑑𝑙 𝑑𝑝𝑒𝑓

  • 2x2:

1 1 1 → 2,1 𝑑𝑝𝑜𝑤. 𝑑𝑝𝑒𝑓, 𝑑𝑒𝑞 = (2,3)

  • 3x3 not possible

1 1 1 ? 1 1 → 𝑂𝑃 2,1 𝑑𝑝𝑜𝑤. 𝑑𝑝𝑒𝑓, 𝑑𝑒𝑞 = (2,3,4)

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The problem addressed here

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The problem addressed here : approach

Add coefficients 𝑠𝑗,𝑘. How many layers 𝑠𝑗,1, … , 𝑠𝑗,𝑙 can be completed, maintaining the s-superregularity? If the layer 𝑠𝐸,1, … , 𝑠𝐸,𝑙 can be completed, maintaining the superregularity, the corresponding code has column distance 2, 3, … , 𝐸 + 2

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Justesen & Hughes (1974) Gluesing-Luersen et. al, «Strongly MDS…», 2006

Previous world records for 𝟑𝒏 ≥ 𝟓

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New constructions : distance 3

Comparison with Wyner-Ash code:

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New constructions: distance 4

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New constructions

Example 1:

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New constructions

𝐼′(2) = 1 1 … 1 … … 𝑏1 𝑏2 … 𝑏𝑙 1 1 … 1 … 𝑐1 𝑐2 … 𝑐𝑙 𝑏1 𝑏2 … 𝑏𝑙 1 1 … 1 Proof: 4

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Proof, distance=4, rate=

𝟑𝒏−𝟐−𝟐 𝟑𝒏−𝟐 construction

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Computer search algorithm

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Computer search algorithm

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Computer search algorithm

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Codes found by computer search

Justesen & Hughes (1974)

Gluesing-Luersen et. al, «Strongly MDS…», 2006

Implicit

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Polynomial notation for convolutional codes

Example 1:

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Codes found by computer search

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Codes found by computer search

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Rareness

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Codes found by computer search

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Codes found by computer search

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Codes found by computer search

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Codes found by computer search

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Codes found by computer search

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Upper bounds

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Conclusions

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Questions? Comments?